Alla Borisyuk
Mathematical Biosciences Institute
Ohio State University
Physiology and mathematical modeling
of the auditory system
Contents
1 Introduction
1.1 Auditory system at a glance . . . . . . . . . . . . . . . . . . . . . . .
1.2 Sound characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Peripheral auditory system
2.1 Outer ear . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Middle ear . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Inner ear. Cochlea. Hair cells. . . . . . . . . . . . . . . . .
2.4 Mathematical modeling of the peripheral auditory system
3 Auditory Nerve (AN)
3.1 AN structure . . . . . . . . . . . .
3.2 Response properties . . . . . . . .
3.3 How is AN activity used by brain?
3.4 Modeling of the auditory nerve . .
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4 Cochlear nuclei
4.1 Basic features of the CN structure . . .
4.2 Innervation by the auditory nerve fibers
4.3 Main CN output targets . . . . . . . . .
4.4 Classifications of cells in the CN . . . .
4.5 Properties of main cell types . . . . . .
4.6 Modeling of the cochlear nuclei . . . . .
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5 Superior olive. Sound localization, Jeffress model
5.1 Medial nucleus of the trapezoid body (MNTB) . .
5.2 Lateral superior olivary nucleus (LSO) . . . . . . .
5.3 Medial superior olivary nucleus (MSO) . . . . . . .
5.4 Sound localization. Coincidence detector model . .
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6 Midbrain
6.1 Cellular organization and physiology of mammalian IC . . . . . . . .
6.2 Modeling of the IPD sensitivity in the inferior colliculus . . . . . . .
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49
7 Thalamus and cortex
58
References
60
2
1
Introduction
It is hard to imagine what the world would be like if it was put on “mute”. For most
people the vast and diverse stream of auditory information is an indispensable part
of the environment perception. Our auditory system, among other things, allows
us to understand speech, appreciate music, and locate sound sources in space. To
be able to perform all perceptual functions the auditory system employs complex
multi-stage processing of auditory information. For example, the sound waves are
separated into frequency bands, which later in the system are often fused to create
perception of pitch, small amplitude and frequency modulations are detected and
analyzed, timing and amplitude differences between the ears are computed, and
everything is combined with information from other sensory systems.
In this chapter we will discuss how the nervous system processes auditory information. We will need to combine some knowledge from anatomy (to know which
structures participate), electrophysiology (that tells us the properties of these structures), and, finally, mathematical modeling (as a powerful tool for studying the
mechanisms of the observed phenomena). In fact, one of the goals of this chapter
is to show examples of types of mathematical modeling that have been used in the
auditory research.
The material in this chapter is based on lectures given by Catherine Carr and
Michael Reed at the Mathematical Biosciences Institute, Ohio State University in
April 2003. It also uses material from the following books: “The Mammalian auditory pathway: neurophysiology” (A.N. Popper, R.R. Fay, eds. [77]), “Fundamentals
of hearing: an introduction” by W.A. Yost [116], ”An introduction to the psychology of hearing” by B.C.J. Moore [67], “From sound to synapse” by C. Daniel Geisler
[26], chapter “Cochlear Nucleus” by E.D. Young and D. Oertel in [90]; and websites:
<http://serous.med.buffalo.edu/hearing/>,
<http://www.neurophys.wisc.edu/aud/training.html>.
1.1
Auditory system at a glance
Let us start by looking at the overall path that the auditory information travels
through the brain. The auditory system differs significantly from the visual and
somatosensory pathways in that there is no large direct pathway from peripheral
receptors to the cortex. Rather, there is significant reorganization and processing
of information at the intermediate stages. We will describe these stages in more
details in the subsequent sections, but for now just outline the overall structure.
Sound waves traveling through the air reach the ears — the outer part of the
peripheral auditory system. Then they are transmitted, mechanically, through the
middle ear to the auditory part of the inner ear — the cochlea. In the cochlea
3
Cortex
Cortex
Thalamus
Thalamus
LSO
DNLL
MSO
MSO
MNTB
LSO
Brainstem
DNLL
Midbrain
IC
IC
MNTB
CN
CN
Figure 1: Schematic of some of the auditory pathways from the ears (at the bottom) to the
cortex. Most of the auditory nuclei appear on both sides of the brain and are shown twice: to
the right and to the left of the (dotted) midline. Abbreviations are as follows: CN – cochlear
nucleus; MNTB – medial nucleus of the trapezoidal body; MSO – medial superior olive; LSO
– lateral superior olive; DNLL – dorsal nucleus of the lateral lemniscus; IC – inferior colliculus.
Excitatory connections are shown with solid lines; inhibitory connections — with dashed.
the mechanical signal is converted into the electrical one by auditory receptors,
namely, hair cells. This transduction process is fundamental in all sensory systems
as external signals of light, chemical compounds or sounds must be “translated”
into the language of the central nervous system, the language of electrical signals.
Next, the signal originating from each cochlea is carried by the auditory nerve
into the brainstem (see Figs. 1, 2). Figure 3 shows a 3-dimensional reconstruction of
the human brainstem. It consists of three main parts: pons, medulla, and midbrain.
The auditory nerve first synapses in the cochlear nuclei inside the medulla. From
the cochlear nuclei, auditory information is split into at least two streams. One
stream projects directly to the auditory part of the midbrain, inferior colliculus
(Fig. 3); while the other goes through another set of nuclei in the medulla, called
4
Figure 2: Location of brainstem in the human brain. Three main parts of the brainstem are
indicated with arrows. (Picture of the brain is from Digital Anatomist Project [20], Department
of Biological Structure, University of Washington, with permission)
superior olivary complex. The first of these pathways is thought to pick up the tiny
differences in acoustic signals, that you need, for example, to differentiate between
similar words. The indirect stream, which is the one that we are mostly going to
consider, originates in the ventral cochlear nucleus. Auditory nerve fibers that bring
information to this stream end with giant hand-like synapses. This tight connection
allows the timing of the signal to be preserved up to a microsecond (which is very
surprising, because the width of action potential, the “unit signal”, is on the order
of milliseconds). This precisely timed information is carried on to the superior olive,
both on the same side of the brain and across midline. In the superior olive the small
differences in the timing and loudness of the sound in each ear are compared, and
from this you can determine the direction the sound is coming from. The superior
olive then projects up to the inferior colliculus.
From the inferior colliculus, both streams of information proceed to the sensory
thalamus (Fig. 1-2). The auditory nucleus of the thalamus is the medial geniculate nucleus. The medial geniculate projects to the auditory cortex, located in the
temporal lobes inside one of the folds of the cortex (Fig. 4).
It is important to notice that many cells at each level of auditory system are
tonotopically organized. This means that in each of the auditory structures it is
5
Inferior Colliculus
Superior Olive
Cochlear Nucleus
Pons
Medulla
Figure 3: Reconstructed view of human brainstem (red), together with thalamus (purple)
and hypothalamus (yellow) (From Digital Anatomist Project [20], Department of Biological
Structure, University of Washington, with permission). Arrows point different parts of the
brainstem, while lines indicate planes within which cochlear nuclei and superior olives are located.
Figure 4: Lobes of the brain (From Digital Anatomist Project [20], Department of Biological
Structure, University of Washington, with permission): frontal lobe (blue), parietal lobe (green),
temporal lobe (purple), occipital lobe (yellow). White arrow points at the fold within which
most of the auditory cortex is located.
6
possible to identify areas in which cells preferentially respond to sounds of certain
frequencies (tones) and the preferred frequencies vary more-or-less continuously with
position of the cell.
Notice also that both pathways starting at the cochlear nuclei are bilateral. The
consequence of this is that localized brain lesions anywhere along the auditory pathway usually have no obvious effect on hearing. Deafness is usually caused by damage
to the middle ear, cochlea, or auditory nerve. In infants the auditory system continues to develop throughout the first year of life. Neurons in the brainstem mature
and many connections are just beginning to form (e.g. between brainstem nuclei,
thalamic nuclei, and auditory cortex). As for other sensory systems, stimulation
during this time is essential for normal development. When hearing is impaired in
early life, the morphology and function of auditory neurons may be affected.
The overall structure of the sub-cortical auditory system and even some anatomical details are similar across many species. A lot of information about the auditory
system has been obtained over the years from experiments, especially anatomical
and electrophysiological, with cats, bats, gerbils, owls, monkeys and many other
animals. Also, a large number of behavioral (psychophysical) studies have been
conducted with animals as well as humans.
1.2
Sound characteristics
Here we describe some of the physical characteristics of the auditory stimuli and
some of the perceptual characteristics that we ascribe to them.
Sounds are pressure waves that transfer energy and momentum from the source
to places around it. One of their natural characteristics is the amount of energy
transmitted by a wave in unit time, called the power of the wave. The power of a
wave is proportional to the square of the frequency, the square of the amplitude and
the wave speed. Often it is convenient to consider the power per unit area of the
wavefront, i.e. the amount of energy transmitted by the wave perpendicularly to its
wavefront in unit time per unit area. This is called the intensity of a wave. The
sound waves can also be characterized by specifying their time-dependent spectra.
The way these physical characteristics of sounds are usually described and quantified
relates to our perception of the sounds.
The spectral properties of many sounds evoke a sensation of pitch. Pitch is
defined as the auditory attribute on the basis of which tones may be ordered on a
musical scale. The pitch of a tone is related to its frequency or periodicity. The
pitch of a periodic sound wave (simple tone) is usually indicated by specifying its
frequency in Hz. The pitch of a complex tone is usually indicated by the frequency
of a simple tone whose pitch would be perceived to be the same. A simple tone
evokes a sensation of pitch only if its frequency is between 20 Hz and 5 kHz, while
7
the spectrum of audible frequencies extends from 20 Hz to 20 kHz.
As a side note: in musical and psychophysical literature there are two different
aspects of the notion of pitch. One is related to the frequency of a sound and is called
pitch height; the other is related to the place in a musical scale and is called pitch
chroma. Pitch height corresponds to the sensation of ’high’ and ’low’. Pitch chroma,
on the other hand, describes the perceptual phenomenon of octave equivalence, by
which two sounds separated by an octave (and thus relatively distant in terms of
pitch height) are nonetheless perceived as being somehow equivalent. Thus pitch
chroma is organized in a circular fashion. Chroma perception is limited to the
frequency range of 50-4000 Hz [122].
The intensity of the sound is commonly quantified on the logarithmic scale in
Bels, or decibels. In this logarithmic scale the intensity I of a sound is quantified
relative to the intensity I0 of a reference sound, and is measured in Bels:
µ
1 Bel = log10
or in decibels (dB):
¶
I
,
I0
µ
¶
I
1
log10
.
10
I0
These expressions give the difference between the intensities in Bels or dB. For
example, if I is hundred times I0 , then the level of I is 2 Bel or 20 dB greater than
that of I0 . If we want to express an absolute intensity I using decibels then we have
to use some standard reference intensity I0 . The reference intensity most commonly
used is 10−12 (Watt/m2 ), which corresponds to a pressure of 20 µPa or about 2·10−10
atmosphere. The sound level of a sound relative to this standard intensity is called
the sound pressure level or SPL of the sound. This standard reference intensity was
chosen because it is very close to the faintest sound of 1 kHz that a human can
detect. This lower limit to the detectable sounds is set by the Brownian motion of
the molecules in the ear. They produce noise input to the auditory receptors, so
the signal has to be strong enough to be detected on top of the noise. Notice that
the sound pressure level can be negative, and that 0 dB sound does not mean the
absence of a sound wave.
The reason for using the logarithmic scale is that when the sound intensity
increases linearly from low to high, it creates the perception that the “loudness”
(i.e. the perceived intensity) increases fast at first and then the increase slows down
as the sound gets louder. Logarithmic scale is also useful because of the great
range of intensities that the auditory system deals with. Dynamic range of human
hearing is approximately 140 dB, ranging from a whisper to a jet engine. To give
some examples, the sound level at an average home is about 40 dB, an average
conversation is about 60 dB, and a loud rock band is about 110 dB.
1 dB =
8
Ossicular chain
Vestibular
system
Cochlea
Inner
ear
Pinna
Eustachian tube
Eardrum
Ear canal
Figure 5: Peripheral auditory system. From <http://www.workplacegroup.net/article-ear-anatomy.htm>,
permission requested.
2
Peripheral auditory system
Peripheral auditory system serves to transfer sounds from the environment to the
brain. It is composed of three main components – outer, middle, and inner ear (Fig.
5).
2.1
Outer ear
The outer ear is the part that we see — the pinna (a structure made of cartilage that we call “the ear” in common speech) and the ear canal. A sound wave
travelling in the air, before it reaches the entrance of the ear canal, must interact
with the head, the torso, and the intrinsic shapes of the pinna. As a result, the
waves can be amplified or attenuated, or their spectrum may be modified. These
changes depend on the specific frequencies of the waves (e.g., they are most significant for waves of frequency above 1.5 kHz), but also on the three-dimensional
position of the sound source. Our brain can use the difference of spectra between
the two ears to determine the position of the original sound source. A typical way
to characterize the pressure that an arbitrary sound produces at the eardrum is
by using the, so called, Head-Related Transfer Functions (HRTFs). These are constructed by placing microphones at the entrance of the subject’s ear canal, and
recording the generated pressure as sounds originate from point sources at various locations. The resulting functions, or their Fourier transforms – HRTF, are
9
surprisingly complicated functions of four variables: three space coordinates and
frequency. Much research has been devoted to measurements, and computations of
the external ear transformations, both for human and animal subjects (see, for
example, [66],[89],[109] for humans, [68] for cats), and databases can be found
on the web, for example at <http://www.ircam.fr/equipes/salles/listen/>,
or <http://interface.cipic.ucdavis.edu/CIL html/CIL HRTF database.htm>.
The HRTFs are used in many engineering applications, most popularly in creation
of realistic acoustic environments in virtual reality settings: games, virtual music
halls, etc. Unfortunately, because of the diversity of the pinnae shapes and head
sizes, HRTFs vary significantly from one person to another. For this reason, each
of the existing models of acoustic fields, based on some average HRTF, is found
adequate by only a relatively small number of listeners.
Having been collected and transformed by the pinnae, the sound waves are directed into the ear canal, where they travel towards the eardrum (also called the
tympanic membrane). The ear canal maintains the proper temperature and humidity for the elasticity of the eardrum. It also contains tiny hairs that filter dust
particles, and special glands that produce earwax for additional protection. The ear
canal can also resonate sound waves and amplify tones in the 3000-4000 Hz range.
2.2
Middle ear
The eardrum (tympanic membrane), and the neighboring cavity with three tiny
bones (ossicular chain) comprise the middle ear (Fig. 5). The sound waves travelling
through the ear canal cause the vibrations of the tympanic membrane, and these
vibrations are further transmitted through the chain of bones towards the inner
ear. For the eardrum to function optimally, it should not be bent either inwards or
outwards in the absence of the sound waves. This means that the air pressure should
be the same in the outer and middle ear. The pressure equalization is achieved
through the eustachian tube — a tube which connects the middle ear cavity with
the back of the throat. Normally this tube is closed, but it opens with swallowing
or shewing, thus working as a pressure equalizer. The middle ear bones (malleus,
incus and stapes) are the smallest bones in the body. They work together as a
lever system, to amplify the force of the vibrations. The malleus is attached to the
tympanic membrane, the stapes enters the oval window of the inner ear, and the
incus lies in between.
2.3
Inner ear. Cochlea. Hair cells.
The inner ear is a series of fluid-filled structures. It has two main parts: a part
that is involved in balance and is called vestibular system, and a second part that is
10
Oval
window
Reissner’s membrane
Wave
propagation
Stapes
Scala Vestibuli
Scala Media
Scala Tympani
Round
window
Cochlear partition
(Tectorial membrane,
hair cells, basilar membrane)
Figure 6: Schematic of a mammalian cochlea, straightened (see text).
responsible for converting sounds from mechanical vibrations into electrical signals,
and is called cochlea. Cochlea, in mammals, is coiled as a snail’s shell (Fig. 5).
It has two membrane-covered openings into the middle ear: the oval window and
the round window, and a membranous sac (containing receptor cells) that separates
them (Fig. 6). The size of the oval window is 15 to 30 times smaller than that of the
eardrum. This size difference produces amplification needed to match impedances
between sound waves in the air and in the cochlear fluid. The principal function of
the membranous cochlear sac is to act as a hydromechanical frequency analyzer.
Here is a rough outline of how it works (see Figure 6). The input from the middle
ear arrives at the oval window and creates pressure fluctuations in the fluid. These
fluctuations travel along the cochlea and eventually are dissipated by the movements
of the large round window, which serves as a pressure release for incompressible
fluids. As the pressure waves travel along the membraneous sac, they permeate
one of the walls (Reissner’s membrane) and cause vibrations in the other (cochlear
partition). The platform of the cochlear partition (basilar membrane) changes in
its mechanical properties along its length from being narrow and stiff at the base
of the cochlea, to being wide and compliant at the apex. Therefore, the lower the
frequency of the tone the further from the oval window the vibration pattern is
located.
Hair cells
Looking at higher resolution inside the cochlear partition (Fig. 7A), there are
auditory sensory cells (inner and outer hair cells) that sit on the basilar membrane
and have their stereocilia sticking out into the fluid and attached to a floppy tectorial
11
B
A
Figure 7: Location and basic structure of hair cells. A: Radial segment of cochlea showing
main components of the cochlear partition. B: Schematic of a hair cell. Bold arrows indicate
that excitation and inhibition of the cell voltage can be produced by deflections of the hair bundle
(see text). Illustration is made by Brook L. Johnson.
membrane. Because the basilar and tectorial membranes attach to the bone at
different points, their motion causes the tectorial membrane to slide across the
basilar membrane, tilting the hair bundle (Fig. 7B). Using electrophysiological
recordings and mechanical stimulation, it has been demonstrated that deflection of
the stereocilia leads to a change of the membrane potential of the cell. Deflection
towards the highest stereocilia leads to a depolarization of the cell. It is also shown
that both the initial change in charge and calcium concentration occur near the tips
of the stereocilia, suggesting that the transduction channels are located at the tips.
Moreover, the calcium entry and charge change happens relatively fast (10 µsec),
which suggests that the transduction channels are gated mechanically rather than
through a second messenger system.
There are two types of auditory sensory cells: one row of inner hair cells (IHCs;
they are called “inner” because they are closer to the bony central core of the twisted
cochlea), and three to five rows of outer hair cells (OHCs). Inner and outer cells
have different functions, and different sets of incoming and outgoing connections.
Each inner hair cell’s output is read out by twenty or so cochlear afferent neurons
12
of type I (for characteristics of types of afferent neurons see below), and each type I
afferent neuron only contacts a single inner hair cell. The output of each outer hair
cell, on the other hand, is read together with many other outer hair cell outputs,
by several type II afferent neurons. All afferent neurons send their axons to the
cochlear nuclei of the brainstem and terminate there. Outer hair cells also receive
descending (efferent) signals from neurons with cell bodies in the brainstem. The
afferent and efferent axons together form the auditory nerve — the communication
channel between the peripheral auditory system and the auditory regions of the
brain.
Inner hair cells convey frequency, intensity and phase of signal. As explained
above, to a first approximation, the frequency is encoded by the identity of the activated inner hair cells, i.e. those inner hair cells that are located at the appropriate
place along the cochlea. The intensity of the signal is encoded by the DC component of the receptor potential; and the timing — by the AC component (see Phase
locking below).
The function of the outer hair cells is not presently known. Two main theories
are that they act as cochlear amplifiers, and that they act to affect the movement of
the tectorial membrane. Another theory is that they function as motors, which alter
the local micromechanics to amplify the sensitivity and frequency selectivity of the
cochlea. It seems that the outer hair cells participate in the mechanical response of
the basilar membrane, because loss of outer hair cells leads to a loss of sensitivity
to soft sounds and decrease in the sharpness of tuning.
2.4
Mathematical modeling of the peripheral auditory system
Mathematical and computational modeling of the peripheral auditory system has
a long history. For mathematical analysis and models of the external and middle
ear see, e.g. review by Rosowski [84]. More recently, there has been a number of
three-dimensional models of the middle ear that use the finite-element method (e.g.
[44, 27, 18, 22, 25, 41, 103]). Some of these models have been successfully used for
clinical applications (e.g. [41, 18]).
An even larger set of studies concentrated on the modeling of the cochlea (for
reviews see, for example, [40, 28]. Early models represented cochlea as one- or twodimensional structure and incorporated only a few elements of the presently known
cochlear mechanics. Early one-dimensional models of the cochlea [23, 76] have assumed that the fluid pressure is constant over a cross-section of the cochlear channel.
The fluid is assumed to be incompressible and inviscid, and the basilar membrane is
modeled as a damped, forced harmonic oscillator with no elastic coupling along its
length. Qualitatively, this model has been shown to capture the basic features of the
13
basilar membrane response. Quantitatively, however, it yields large discrepancies
with measurement results [120]. Two-dimensional models by Ranke [78] and Zwislocki [121] make similar assumptions on the cochlear fluid and the basilar membrane.
Ranke’s model uses a deep water approximation, while Zwislocki used the shallow
water theory in his model. These models were further developed in [3, 4, 53, 92]
and in other works. Other two-dimensional models incorporate more sophisticated
representations of the basilar membrane using, for example, elastic beam and plate
theory [10, 17, 36, 37, 43, 99]. Three-dimensional models were considered by Steele
and Taber [100] and de Boer [19], who used asymptotic methods and computations
and obtained an improved fit of the experimental data. Their work seems to indicate
that geometry may play a significant role in the problem. In particular, the effect
of the spiral coiling of the cochlea on the wave dynamics remains unresolved; see
[101, 105, 56, 60].
With the development of more powerful computers it became possible to construct more detailed computational models of the cochlea. A two-dimensional computational model of the cochlea was constructed by Beyer [7]. In this model the
cochlea is a flat rectangular strip divided into two equal halves by a line which
represents the basilar membrane. The fluid is modelled by the full Navier-Stokes
equations with a viscosity term, but elastic coupling along the basilar membrane
is not incorporated. Beyer has used a modification of Peskin’s immersed boundary
method, originally developed for modeling the fluid dynamics of the heart [75] .
Several three-dimensional computational models have been reported, such as Kolston’s model [45], intended to simulate the micro-mechanics of the cochlear partition
in the linear regime (i.e., near the threshold of hearing), Parthasarati, Grosh and
Nuttal’s hybrid analytical-computational model using WKB approximations and
finite-element methods, and Givelberg and Bunn’s model [28] using the immersed
boundary method in a three-dimensional setting.
We will consider as an example the analysis of a two-dimensional model with
fluid viscosity, by Peskin [73, 74]. In this model cochlea is represented by a plane
and the basilar membrane by an infinite line dividing the plane into two halves.
The fluid in this model satisfies the Navier-Stokes equations with the non-linearities
dropped. The distinctive feature of this model is that the location of wave peak
for a particular sound frequency is strongly influenced by fluid viscosity and the
negative friction of the basilar membrane. This model was studied with asymptotic
and numerical methods in [54, 55]. In the example that we present here, the zeroth
order approximation of the solution is found by WKB method. (The WKB method
was first used for this cochlear model by Neu and Keller [69] in the case of zero
membrane friction).
Problem setup: Basilar membrane at rest is situated along the x-axis. The
deviation of the basilar membrane from the axis will be denoted by h(x, t). Vector
14
(u, v) is the velocity of the fluid, and p is the pressure of the fluid (these are functions
of (x, y, t)).
Assumption 1: the fluid satisfies the Navier-Stokes equations with the nonlinearities left out, i.e.
for y 6= 0 :
Ã
!
∂2u ∂2u
∂u ∂p
+ 2 ,
ρ
+
=µ
∂t
∂x
∂x2
∂y
Ã
∂v ∂p
∂2v
∂2v
ρ
+
+
=µ
∂t
∂y
∂x2 ∂y 2
(1)
!
,
(2)
∂u ∂v
+
= 0.
∂x ∂y
(3)
In these equations, ρ is the density and µ is the viscosity of the fluid. The last
equation represents the incompressibility of the fluid.
Notice that there is no explicit forcing term in this model. Yet, there are bounded
wave solutions that move in the direction of the increasing x as if there is a source
of vibration at x = −∞.
Assumption 2:
a) the basilar membrane (located at y = 0) has zero mass;
b) there is no coupling along the membrane;
c) each point of the membrane feels a restoring force that is proportional to the
displacement and to the compliance (flexibility) of the membrane, the latter given
by eλx (to represent the fact that the actual membrane is more narrow and more
flexible at the far end);
d) the membrane possesses an active mechanism (mechanical amplifier, parameter
β below is negative), i.e.
for y = 0 :
u(x, 0, t) = 0,
v(x, 0, t) =
(4)
∂h
(x, t),
dt
µ
p(x, 0− , t) − p(x, 0+ , t) = s0 e−λx h + β
(5)
¶
∂h
(x, t).
∂t
(6)
Note that the boundary conditions are applied at the rest position of the membrane
y = 0, not at its instantaneous position h(x, t). This, as well as the above assumptions is justified by the small displacements of the membrane and the fluid particles
in the cochlea. Notice also that there is an unknown function h(x, t) in the boundary
conditions. Finding this function such that the rest of the system has a bounded
15
solution (u, v, p) is part of the problem. In addition, the problem is complicated by
presence of the stiffness term. The parameter λ was measured (for dead cochlea) by
von Békésey: λ−1 ≈ 0.7 cm, i.e. over the length of the human cochlea (length ≈ 3.5
cm) the stiffness more than doubles.
Because of the symmetry of the system, we look for solutions that satisfy
p(x, y, t) = −p(x, −y, t),
u(x, y, t) = −u(x, −y, t),
v(x, y, t) = v(x, −y, t).
Then we can restrict the system of equations (1-3) to y < 0 and, using notation
y = 0 instead of y = 0− , re-write the boundary conditions (4-6) as
u(x, 0, t) = 0,
(7)
∂h
(x, t),
dt
µ
¶
∂h
−λx
2p(x, 0, t) = s0 e
h+β
(x, t).
∂t
We also impose the condition
v(x, 0, t) =
(8)
(9)
(u, v, p) → 0 as y → −∞.
Solution plan: We want to determine the solution that represents the steady
state response of the cochlea to a pure tone. We will look for this solution as a
small perturbation from the time-periodic function of the same frequency as the
tone (and thus introduce a small parameter in the system). Further, we will use
asymptotic expansion in the small parameter and by solving the zeroth order system
of equations we will find an approximation of the solution.
We use a change of variables
 


u
U
¢
¡
Φ(x−xε )
 v  (x, y, t, ²) =  V  (x − xε , y/ε, ε)ei ωt+ ε
,
p
P
¡
h(x, t, ε) = H(x − xε , ε)ei
ωt+
Φ(x−xε )
ε
¢
,
where Φ is the local spatial frequency, ω is the given frequency of the external pure
tone stimulus (radians/second), functions U, V, P, H, Φ are complex-valued, and the
parameters ε and xε will be chosen later on. We set
X = x − xε ,
16
Y = y/ε,
and
ξ(X) = Φ0 (X) =
∂Φ
(X).
∂X
In terms of the new variables:
¡
¢
Φ(x−xε )
∂u
= iω · U · ei ωt+ ε
,
∂t
·
¸
¢
¡
Φ(x−xε )
∂u
∂U
i
,
=
+ ξ(X)U ei ωt+ ε
∂x
∂X
ε
¢
ε)
∂u
1 ∂U i¡ωt+ Φ(x−x
ε
=
e
,
∂y
ε ∂y
∆u = ∆U =
h
=
∂2U
∂X 2
+
i
ε
³
´
∂U
ξ 0 U + 2ξ ∂X
+
³
1
ε2
∂2U
∂Y 2
− ξ2U
´i ¡
ei
ωt+
Φ(x−xε )
ε
¢
.
Then the equations (1-3) can be rewritten as:
for Y < 0 :
¶
µ
∂
ξ
P
(iωρ − µ∆)U + i +
ε ∂x
1 ∂P
(iωρ − µ∆)V +
ε ∂Y
µ
¶
ξ
∂
1 ∂V
i +
U+
ε ∂x
ε ∂Y
= 0,
(10)
= 0,
(11)
= 0,
(12)
with boundary conditions
for Y = 0 :
U (X, 0, ε)
=
0,
V (X, 0, ε)
=
iωH(X, 0, ε),
2P (X, 0, ε)
=
s0 (1 + iωβ)e−λxε e−λx H(X, 0, ε),
(U, V, P ) → 0
as
Y → −∞.
Assumption 3. The functions U, V, P, H have the expansion
U = U0 + εU1 + · · · ,
17
V = V0 + εV1 + · · · ,
P = ε(P0 + εP1 + · · ·),
H = H0 + εH1 + · · · .
We now choose
ε2 =
µλ2
,
ρω
e−λxε = ε.
This choice of ε makes it indeed a small parameter for realistic values of other
quantities. For example, if the fluid has characteristics of water (ρ = 1 g/cm3 ,
µ = .02 g/(cm· s)), for a 600 Hz tone (ω = 2π · 600/s) and 1/λ=.7 cm, ε ≈ .003.
If we substitute the expansions of U, V, P, H into the equations (10-12) and collect
terms with matching powers of ε, we find at the zeroth order:
for Y < 0 :
Ã
Ã
1
ρω i − 2
λ
Ã
∂2
−ξ +
∂Y 2
!!
2
1
ρω i − 2
λ
Ã
∂2
−ξ 2 +
∂Y 2
iξU0 +
U0 + iξP0 = 0,
!!
V0 +
∂P0
= 0,
∂Y
∂V0
= 0,
∂Y
and for Y = 0 :
U0 (X, 0) = 0,
V0 (X, 0) = iωH0 (X),
2P0 (X, 0) = s0 (1 + iωβ)e−λx H0 (X),
(U0 , V0 , P0 ) → 0 as Y → −∞.
Next, at the first order
Ã
ρω i −
1
λ2
Ã
Ã
−ξ 2 +
1
ρω i − 2
λ
Ã
∂2
∂Y 2
for Y < 0 :
!!
µ
U1 + iξP1 =
∂2
−ξ 2 +
∂Y 2
!!
iξU1 +
ρω
∂
i ξ 0 + 2ξ
λ2
∂X
µ
¶
U0 −
∂P1
ρω
∂
V1 +
= 2 i ξ 0 + 2ξ
∂Y
λ
∂X
∂V1
∂
=−
U0 ,
∂Y
∂X
18
∂
P0 ,
∂X
¶
V0 ,
and for Y = 0 :
U1 (X, 0) = 0,
V1 (X, 0) = iωH1 (X),
2P1 (X, 0) = s0 (1 + iωβ)e−λX H1 (X),
(U1 , V1 , P1 ) → 0 as Y → −∞.
Consider the zeroth order equations. For each fixed x, the functions P0 , U0 , V0 satisfy
an ODE system whose solution is given by
√
2
P0 (X, Y ) = P0 (X, 0)e ξ Y ,
¶
µ √
√
iξ
2
2
2
e ξ Y − e ξ +iλ Y ,
iωρ
 √

√
2
2
2
ξ2  e ξ Y
e ξ +iλ Y 
p
V0 (X, Y ) = −P0 (X, 0)
−p 2
,
iωρ
ξ2
ξ + iλ2
U0 (X, Y ) = −P0 (X, 0)
√
where
denotes the root with positive real part.
Next, the zeroth order equations that contain H0 give two different formulae for
H0 in terms of P0 (X, 0):
V0 (X, 0)
ξ2
H0 (X) =
= P0 (X, 0) 2
iω
ω ρ
and
H0 (X) = P0 (X, 0)
Ã
1
1
p −p
ξ2
ξ 2 + iλ2
2eλX
.
s0 (1 + iωβ)
!
(13)
(14)
We have just expressed our zeroth order solution U0 (X, Y ), V0 (X, Y ), P0 (X, Y ),
H0 (X) in terms of P0 (X, 0) and ξ(X). Now there are two steps remaining. Step 1
is to find a function ξ(X) such that both equations for H0 (X) are satisfied. Step 2
is to find P0 (X, 0).
Step 1. For ξ to satisfy both (13) and (14) we need to have
Ã
Q(X) = ξ 2
1
1
p −p
2
2
ξ
ξ + iλ2
where
Q(X) =
2ρω 2 eλX
.
s0 (1 + iωβ)
19
!
,
(15)
The condition (15) is called a dispersion relation. It implicitly defines the local
spatial frequency ξ in terms of X and ω. Equation (15) may have multiple solutions,
for example, if ξ is a solution, then
p −ξ is also a solution (a wave going in the opposite
direction). Using notation η = ξ 2 we can transform (15) into
1
iλ2 2
1
η 3 − (Q +
)η + iλ2 η − iλ2 Q = 0.
2
Q
2
But not every zero of this cubic satisfies the dispersion relation. First of all the real
part of η has to be non-negative. Second, by definition of Q: Q < η, therefore only
the zeros of the cubic such that
Q=η−p
η2
η 2 + iλ2
satisfy the original equation. For every root of cubic that satisfies these two conditions we can then take ξ(X) = ±η.
Step 2. Now we need to find P0 (X, 0). If we multiply the first order equations
by functions −U0 , V0 , P0 , respectively (which represent solution to the zeroth order
equations with ξ replaced by −ξ), and integrate each equation over Y ∈ (−∞, 0),
then we obtain after integration by parts and combining non-vanishing terms
∂
∂X
i.e.
Z 0 ·
iωρξ
−∞
Z 0 ·
iωρξ
−∞
¸
(V02
λ2
(V02
λ2
−
U02 )
+ U0 P0 dY = 0,
¸
−
U02 )
+ U0 P0 dY = C0 ,
where C0 is a constant independent of X. Notice that all terms containing U1 , V1
and P1 have disappeared because of the vanishing factors in front of them. Now
substitute U0 , V0 and P0 by their expressions in terms of P (X, 0), and integrate.
We get
p
p
2ωρC0 ( ξ 2 + iλ2 )3 ξ 2
2
p
p
p
(P0 (X, 0) ) = p 2
,
ξ( ξ + iλ2 − ξ 2 )( ξ 2 + iλ2 ξ 2 − iλ2 )
which gives us the solution.
3
3.1
Auditory Nerve (AN)
AN structure
The auditory nerve is a collection of axons connecting the peripheral auditory system
and the auditory areas of the brain. It is made up of approximately 30,000 to 55,000
20
nerve fibers, depending on species. About 95% of them are afferent, projecting from
cell bodies in the cochlea to cochlear nuclei in the brainstem, and the rest are efferent,
coming to cochlea from cells in the olivary complex (also part of the brainstem, see
below). The afferent neurons are divided into Type I and Type II, based on their
morphology: type I cells are large, have bipolar shape, their axons are large and
myelinated, and, therefore, fast; type II cells are smaller, have different shape and
non-myelinated axons. In addition, type I fibers innervate inner hair cells in manyto-one fashion, and type II fibers innervate many-to-many outer hair cells. Very
little is known about the functional properties of type II afferents, partially because
type I fibers are much easier for physiologist to record from, due to their large size
and number. The role of efferent neurons in modifying the auditory input is also
not yet clear. So, the rest of this text will focus on type I afferents.
3.2
Response properties
Spontaneous rates
In mammals, afferents can be divided into low, medium and high spontaneous
rate fibers. The spontaneous rate (firing rate in the absence of stimuli) is determined
by pattern of hair cell innervation, although the mechanisms are unclear. It could
be that either smaller size of low spontaneous rate fibers makes them less excitable,
or less transmitter is released from the hair cells to low spontaneous rate fibers.
The variety of available fiber sensitivity provides a way of encoding wide range of
intensities in auditory nerve (see Intensity sensitivity below).
Thresholds
The threshold is defined as minimal intensity of the stimulus that increases firing
rate above the spontaneous level. Note that this is not a real threshold because, for
example, for very low frequency fibers, sound first synchronizes spikes before there
is ever an increase in rate. Usually the threshold is about 1 dB in mammals.
Latencies
Auditory fibers are also characterized by latencies in their responses. The latencies can be measured using brief stimuli, for example, clicks. Fibers tuned to
different frequencies respond to a click at different latencies. Most of the delay originates from travel time of wave along basilar membrane. High frequency areas on
the membrane are stimulated first, i.e. high frequency cells have shorter latencies.
Frequency tuning
If we fix the intensity of the stimulus at any particular level and look at the firing
rate of the given fiber at different frequencies, we find that the response function is
21
usually single-peaked. This means that the cell has a preferred range of frequencies,
inherited from its innervation of cochlea.
A typical way to characterize the basic properties of an auditory cell, is to probe
its responses to a variety of pure tones (stimuli with only one frequency component
and constant intensity). These responses are often summarized by marking the
areas of intensity and frequency that produce a change in the firing rate of the cell
(response areas). The border of this area is called a tuning curve of the cell. For
auditory nerve fibers (Fig. 8) these areas usually have triangular shape, pointing
down, and most of the area corresponds to an increase in firing rate (excitatory).
The frequency at which the tip of the response area is located, i.e. the frequency
at which the cell is most sensitive, is called the cell’s characteristic frequency (CF)
or best frequency (BF). The CF of each cochlear afferent is well-defined and it
provides accurate information about the position on the cochlea of the hair cell
that the afferent innervates. Approximately, the linear distance on the cochlea
is proportional to the logarithm of CF. The shapes of the response areas change
systematically with CF. The response areas of high-CF fibers have very steep highfrequency slope, and elongated tip (Fig. 8). Response areas of lower-CF afferents
are relatively broader and more symmetrical. Notice that often relative bandwidth
(normalized to CF), rather than absolute bandwidth, is used to characterize the
sharpness of tuning; and relative bandwidth increases with CF. A common measure
of frequency tuning is the Q10 , defined as the CF divided by the bandwidth at 10
dB above CF threshold. Q indexes also get higher with increasing CF.
Intensity tuning
Next property of the auditory nerve fibers, that we will be discussing, is the
intensity tuning. Let us keep the frequency constant and consider responses, in
terms of firing rate, at different intensity levels (rate-intensity function). This function is generally monotonically increasing over 40-50 dB above threshold, and then
saturates. The range over which this function is increasing is termed the neuron’s
dynamic range. Maximum dynamic range is usually at CF. Notice that the dynamic
range of individual cells (40 dB) is 107 times smaller than the total range of human hearing (110 dB). To build such a wide range out of small-range elements, the
auditory system makes use of the diversity in sensitivity of individual cells. The
threshold levels of less sensitive fibers are situated at the upper end of the range of
the more sensitive ones. In this way one set of neurons is just beginning to fire above
their spontaneous rate when the other group is beginning to fire at their maximal
rate. Additionally, at very high sound levels, the frequency tuning properties of the
cochlea break down. Thus, we are able to tell that a 102 dB sound is louder than a
100 dB sound, but it is hard for us to determine whether it has a different frequency.
22
SPL, dB
Frequency, kHz
Figure 8: Examples of frequency-threshold tuning curves for chinchilla auditory nerve fibers.
Area above each curve is the response area of the given fiber. Reprinted by permission of VCH
Publishers, Inc. from [86].
Phase Locking (transmission of timing information)
Phase-locking of one (potentially stochastic) process with respect to another
(often periodic) means that the events of the former preferentially occur at certain
phases of the latter; in other words, that there is a constant phase shift between the
two.
In the presence of a pure tone (periodic) stimulus, due to cochlear structure,
inner hair cells are stimulated periodically by the pressure waves. If the frequency is
low enough, then the depolarizations of the hair cell are also periodic and occur at
certain phases of the stimulus cycle. This, in turn, generates a phase-locked release
of neurotransmitter (that carries the neuronal signal to other cells) and, ultimately,
leads to spikes in the auditory nerve that are also phase-locked to the stimulus cycle.
To observe or measure the phase-locking, people often build a histogram of
phases of the recorded spikes within a period of the stimulus (Fig. 9). To have a
meaningful resolution, given that stimuli frequencies can be quite high, bin widths
have to be in the microsecond range. Phase-locking is equivalent to spikes accumulating at some parts of the periodogram, i.e. forming a peak.
A typical way to quantify phase-locking is by computing vector strength of the
period-histogram [30]. Vector strength of a periodic function is the norm of the first
complex Fourier component, normalized by the norm of the zeroth Fourier component. Normalization is included to remove dependence on the mean firing rate. This
quantity can vary between 0 and 1, with 0 indicating low phase locking (for example,
23
Number of spikes in a bin
300 Hz
N=368
VS=0.844
400 Hz
N=391
VS=0.807
701 Hz
N=341
VS=0.604
1101 Hz
N=321
VS=0.493
500 Hz
N=365
VS=0.783
600 Hz
N=366
VS=0.656
801 Hz
N=323
VS=0.573
901 Hz
N=254
VS=0.498
1003 Hz
N=259
VS=0.434
1202 Hz
N=299
VS=0.430
1304 Hz
N=236
VS=0.412
1403 Hz
N=207
VS=0.388
One period of stimulus time, msec
Figure 9: Periodograms of responses of an auditory nerve fiber. Each panel shows response to
one stimulus frequency, and horizontal axis spans one stimulus period. Above the histogram is
the stimulus frequency, total number of spikes (N) and the vector strength (VS). Reprinted by
permission of Acoustical Society of America from [5].
in case of uniform distribution of spikes), and 1 indicating high phase locking (when
all spikes fall in the same bin). Figure 9 shows an example of periodograms and
vector strengths of an auditory fiber in response to stimuli of different frequencies.
Phase-locking generally occurs for stimuli with frequencies up to 4-6 kHz. Even
high frequency fibers, when stimulated with low frequencies at intensities 10-20 dB
below their rate threshold, will show phase locking of spikes, without increase in
firing rate [39]. Decline of phase-locking with increase in frequency originates in the
parallel reduction of the AC component of the inner hair cell response.
Phase-locking plays an important role in our ability to localize sounds, particularly at low frequencies, at which interaction of sound waves with the body and the
pinna is reduced. We are able to tell which direction a sound is coming from (its
azimuth), based on time delay between when it reaches our right and left ears. The
idea is that the sound, originating on one side of the head, must follow a longer path
and consequently takes a longer time traveling to one ear than to the other. Then,
24
due to the phase locking, the time delay is translated into the phase shift between
spike trains, originating in the different sides of the brain. These delays are detected
in some parts of the auditory pathway superior olivary complex, with precision as
small as 20 microseconds. We will come back to the issue of phase-locking and sound
localization when we talk about superior olivary complex below.
3.3
How is AN activity used by brain?
One of the basic questions one can ask about how the AN signal is decoded is the
following: how does a neuron differentiate between change in rate due to change in
intensity vs change in rate due to change in frequency? While the complete answer
is not yet clear, it seems that the key is to look not at the responses of the single
cell, but at the distribution of the responses across a population. The basic view is
that the loudness can be carried by the total level of the population activity, and
the frequency — by the position of the activity peak within the population (place
principle) and/or by the temporal structure of the spike trains (using phase-locking
— volley principle).
In this simple view, one considers complex sounds as made up of sinusoids
(Fourier components). This linear approximation works well (at normal acoustic
pressures) for the outer ear, less well for middle ear and only partially for the inner
ear. The inner ear, anatomically, breaks sound into bands of frequencies, that persist for many stages of processing. How the information from these different bands
is put back together is unknown.
In reality things are not linear. We will give two examples of non-linearities. The
first example shows that the responses of AN fibers to tones depend on the spectral
and temporal context. We illustrate this by describing a two-tone suppression:
response of AN fiber to a given tone can be suppressed by a presence of another
tone of a given frequency. The second example illustrates the non-linearity in a
phenomenon of “hearing the missing fundamental”: tone combinations can create
perception of frequencies that are not really present in the signal. Now we describe
these examples in more detail.
Example 1. Two-tone suppression occurs in recordings from cochlea or afferents.
It is defined as reduction in response to one tone in presence of a second tone. It
depends upon levels and frequencies of the two tones. To illustrate this phenomenon,
one can map response area of a cell (Fig. 10), then choose a test tone near CF
(triangle in Fig. 10), and mark by shading the tonal stimuli that suppress responses
to the test tone. Regions of overlap of the shading with the response area show
that some tones that are excitatory by themselves, can suppress responses to CF
tones. Latency for suppression is as short as initial response, i.e. it is not caused
by efferents. It is shown that major component of the suppression originates from
25
SPL, dB
Frequency, kHz
Figure 10: Example of two-tone suppression. Tuning curve (black line), showing
the lower border of the response area of an auditory nerve fiber. Response
to test tone (triangle) can be suppressed by any of the tones in the shaded
area. Recording from chinchilla auditory nerve fiber, reprinted from [33] with
permission from Elsevier.
basilar membrane motion.
Example 2. The second example of non-linearity is that in some specific types
of experiments it is possible to hear a sound of certain frequency even though the
ear is being stimulated by several other frequencies. This phenomenon is sometimes
referred to as “hearing the missing fundamental”. For example, three frequencies
are played simultaneously F1=4000 Hz, F2=6000 Hz, F3=8000 Hz. Rather than
hearing these three frequencies, the listener actually hears a 2000 Hz sound. It
has been shown that the region of the cochlea with its threshold tuned to 2000 Hz
do not fire any faster, while the neurons ”tuned” to 4000 Hz, 6000 Hz and 8000
Hz are firing well above their spontaneous rate. To account for this paradox some
scientist have proposed that frequency discrimination is not done based on basilar
membrane resonance, but on timing information. Due to phase locking and because
all frequencies present are multiples of 2000 Hz, multiple auditory nerve fibers will
fire simultaneously at every cycle of the 2000 Hz oscillation. It is hypothesized that
if only this timing information is used to interpolate the frequency content of the
sound, the brain can be tricked into hearing the missing fundamental.
26
3.4
Modeling of the auditory nerve
Mathematical modeling of the responses of the auditory nerve fibers has mostly
been phenomenological. The models aimed to described as accurately as possible the experimentally known characteristics of fibers’ responses. For example, to
match the responses to single simple tones or clicks and to temporal combinations
of different stimuli [16, 104, 119], or to determine the lower thresholds of the singlepulse responses [111]. Prevalence of this type of modeling is due to the fact that
the biophysics of the hair cells, where auditory nerve fibers originate, has not been
worked out in sufficient detail to allow construction of more biophysically realistic
models. However, the auditory nerve fiber modeling has recently become one of the
examples of how just how much theoretical studies can contribute to the field of
neuroscience. Recent models of Heinz et al. [34, 35] provide a link between auditory fiber physiology and human psychophysics. They find how results equivalent
to human psychophysical performance can be extracted from accurate models of the
fiber responses. In essence, this demonstrates the “code” in auditory nerve fibers
via which the auditory system extracts all necessary information.
4
Cochlear nuclei
As fibers of the auditory nerve enter the cochlear nuclei (CN), they branch to form
multiple parallel representations of the environment. This allows parallel computations of different sound features. For example, sound localization and identification
of a sound are performed in parallel. This separation of processing pathways exists
both in the CN and further upstream in the brainstem.
Some information carried by auditory nerve fibers is relayed with great fidelity
by specific neurons over specific pathways to higher centers of the brain. Other CN
neurons modify the incoming spike trains substantially, with only certain elements
of the input signal extracted prior to transmission to the next station in the auditory
pathway.
One of the features that is inherited in CN from the auditory nerve is tonotopy
(continuous variation of the characteristic frequency with the position of the cell,
as on basilar membrane). But, interestingly, tonotopic projections are not pointto-point. Each characteristic frequency point on basilar membrane projects to an
iso-frequency plane across the extent of the cochlear nucleus. Thus cochlear place
representation is expanded into a second dimension in brain. These tonotopic sheets
are preserved in projections all the way to cortex. We contrast this with the visual
and somatosensory systems, where maps reflect a location of stimulus in space and
the representations are point to point.
In this section we will first describe general structure of the CN and outline its
27
Figure 11: Schematic of cell type location in CN. Main parts of the CN are marked by color
(green is dorsal cochlear nucleus (DCN), yellow is posteroventral cochlear nucleus (PVCN) and
blue is anteroventral cochlear nucleus (AVCN)). Cell types are marked by symbols. Illustration
is made by Brook L. Johnson.
input and output streams. Then we will describe in more details the properties of
cells that make information processing in the CN possible.
4.1
Basic features of the CN structure
As we mentioned above, there are parallel streams of information processing in the
cochlear nuclei. Thus, the CN is naturally subdivided into areas that house cells
with different specialized properties, receive different patterns of inputs and project
to different targets. The main subdivision structures are the ventral cochlear nucleus
(VCN) and the dorsal cochlear nucleus (DCN) (Fig. 11).
The VCN contains four types of principal cells: globular bushy cells, spherical
bushy cells, multipolar cells and octopus cells. We will describe their properties
below, and just mention now that they play a very important role in processing
different sorts of timing information about the stimuli.
Unlike VCN, DCN is layered and has interneurons. The outermost layer, called
the superficial or molecular layer, contains cell bodies and axons of several types
of small interneurons. The second layer, called he pyramidal cell layer, has the cell
bodies of pyramidal cells, the most numerous of the DCN cell type, and cartwheel
and granule cells. The deep layer contains the axons of auditory nerve fibers as well
as giant cells and vertical cells.
28
4.2
Innervation by the auditory nerve fibers
The main external input to the CN comes from the auditory nerve fibers. There
is also input to granule cells that brings multimodal information from widespread
regions of the brain, including somatosensory, vestibular and motor regions, but we
will not consider it here.
As a reminder, in mammals there are two types of the auditory nerve fibers.
Type I fibers innervate one inner hair cell each, the fibers are thick and myelinated,
and they constitute 90-95% of all fibers. Type II fibers innervate outer hairs cells,
they are thinner and unmyelinated.
Type I fibers in the cochlear nuclei form two branches: the ascending branch
goes to the anteroventral region of the cochlear nucleus (AVCN) and the descending
branch to the posteroventral region (PVCN) and parts of DCN. Type II fibers project
to DCN, but because it is not clear if they carry auditory information, type II fibers
are, once again, not considered here.
Auditory nerve fibers make synapses to all cell types in the cochlear nuclei,
except in the molecular layer of the DCN and in the granule cells region. The
innervation is tonotopic within each principal cell type in the VCN and in the deep
layer of DCN.
Nerves form different types of terminals onto different cell types and/or different
cochlear nucleus divisions. The terminals range from small to large endbulbs. The
largest are the endbulbs of Held (calices of Held) onto the bushy cells. Each such
terminal contains hundreds of synapses. This allows to inject a lot of current into
postsynaptic cell every time the pre-synaptic signal arrives. Perhaps only one endbulb is needed to fire a cell, and the transmission through these connections is very
fast and reliable. Other CN cell types receive input from auditory nerves through
more varicose or bouton-like terminals, located at their dendrites. In these cases
more integration is required to fire a cell.
All auditory nerve synapses use glutamate as the neurotransmitter, and are depolarizing (excitatory). Often postsynaptic cells have specialized “fast” receptors
(of AMPA type), important in mediating precise coding. Interestingly, in young animals glutamate receptors in the VCN are dominated by very slow NMDA receptors
and are replaced by AMPA receptors in the course of development. The reason for
this change is not presently clear.
Another important feature that is modified with age is synaptic plasticity. In
many neuronal systems the strength of synapses are modified, both on the short
(during a single stimulus response) and on the long time scale. In contrast, synaptic
transmission by adult auditory nerve fibers shows little plasticity. Synaptic depression (decrease in synaptic efficacy on time scale of 100 msec) is prominent in young
animals and decreases with age.
29
to IC
CN
LSO
MSO
LNTB
MNTB
MSO
LSO
MNTB
LNTB
Pyr
Gi
M
GBC
SBC
CN
VNLL
VNLL
Pyr
Gi
M
GBC
SBC
Figure 12: Main ascending connections from cochlear nucleus (CN). Main CN cell types:
globular bushy cells (GBC), spherical bushy cells (SBC), multipolar (M), giant (Gi), pyramidal
(Pyr) make connections to medial nucleus of the trapezoidal body (MNTB), lateral nucleus of
the trapezoidal body (LNTB), medial superior olive (MSO), lateral superior olive (LSO), ventral
nucleus of lateral lemniscus (VNLL)and inferior colliculus (IC). All connections are symmetric
with respect to midline (dotted line), but not all are shown. Excitatory connections are shown
with solid lines and inhibitory with dashed.
4.3
Main CN output targets
Figure 12 shows the two major fiber bundles that leave the cochlear nucleus in
relation to other structures in the brainstem. It also shows some of the auditory
neuronal circuits of the brainstem. At the output of the cochlear nucleus, spherical
and globular bushy cells (SBC and GBC) project to the medial (MSO) and lateral
(LSO) superior olivary nuclei and the trapezoid body (MNTB and LNTB). The MSO
and LSO compare input from the two ears and perform the initial computations
necessary for sound localization in the horizontal plane (see next section). Excitatory
inputs to the superior olive come mostly from the spherical bushy cells and inhibitory
inputs come from the globular bushy cells via the inhibitory interneurons in the
trapezoid body.
Multipolar, giant, and pyramidal cells project directly to the inferior colliculus,
where all ascending auditory pathways converge (see section 6). The octopus cells
project to the superior paraolivary nucleus and to the ventral nucleus of the lateral
lemniscus (VNLL). The granule cells axons project to the molecular layer of the
DCN, where they form parallel fibers. These parallel fibers run orthogonal to the
auditory nerve fibers and cross isofrequency layers. The input from these two sources
is combined by the principal cells of the DCN – the pyramidal cells.
30
4.4
Classifications of cells in the CN
In the current prevalent view, each cochlear nucleus cell type corresponds to a
unique pattern of response to sound; this is consistent with the idea that each type
is involved in a different aspect of the analysis of the information in the auditory
nerve. The diversity of these patterns can be accounted for by three features that
vary among the principal cell types: (1) the pattern of the innervation of the cell by
ANFs, (2) the electrical properties of the cells, and (3) the interneuronal circuitry
associated with the cell.
To study the correlation between physiological and anatomical properties of the
cells experimentally, it is necessary to make intracellular recordings of responses to
various current injections and then fill the cells with a dye to image their shape.
These experiments are hard to perform and while there seems to be evidence that
the correlation between structure and function in these cells is strong, this evidence
is not conclusive.
Physiological types
Three major types of VCN cells (bushy, multipolar and octopus) probably correspond to different physiological types: primary-like, chopper and onset.
Primary-like (spherical bushy) and primary-like with notch (globular bushy)
responses are very much like auditory nerve. Initial high burst of spikes (>1,000
Hz) followed by decline to maximum rate of no more than 250 Hz. Pause (notch) is
mostly due to refractory period of cell.
Choppers are major response type in PVCN, also found in other divisions. In
response to high frequency tones, they discharge regularly, independent of stimulus
frequency and phase with firing rates sustained at up to 200-500 Hz. These appear
to be multipolar cells.
Onset responses have spikes at the onset of stimulus, and then fewer or none.
Standard deviation of first spike latency is very small, about 100 µsec. Those few
spikes that are produced during the ongoing stimulus phase lock to low frequency
tones. These cells are located mostly in PVCN. Some of them are octopus cells and
others are large multipolars. The properties of these responses (precise onset and
brief transient activation) allow them to play an important role in temporal coding.
DCN cells exhibit wide variety of response types. Their functions probably are
related to pinna movement and echo suppression (which allows you to not hear
yourself many times when you speak in a small room). In addition, there is also
somatosensory input through granule cells that has to be combined with the auditory
one.
In vitro physiology
31
To complicate things further, slice physiology in VCN provides additional view
of types. In these studies intracellular recordings are made from isolated cells. It is
possible to either control the current flowing across the cell membrane and measure
the induced voltage (current clamp), or to fix the membrane voltage and to record
the resulting trans-membrane current (voltage clamp). The recordings reveal two
major physiological response types. Type 1 fires regular train of action potentials in
response to depolarization. Also, it has relatively linear current-voltage curve. Type
2 fires just one, or a few, action potentials in response to ongoing depolarization. In
addition it has very nonlinear current-voltage curve, with zero (reversal potential)
at about -70 mV and much steeper slope in depolarizing than in hyperpolarizing
direction.
Differences between types 1 and 2 are largely due to change in potassium currents. Type 2 cells have low threshold potassium current that is partially activates
near rest values of voltage and strongly activates when voltage rises, repolarizing
the membrane and halting the response. Many type 1 cells are multipolar and many
type 2 cells are bushy (see more on this below, under Bushy and Multipolar cells).
4.5
Properties of main cell types
Bushy cells
Bushy cells have short (<200 µm), bushy dendritic trees. Their synaptic input
is located mainly on the soma, with few synapses on their dendrites. Two subtypes
of bushy cells are recognized as spherical and globular.
Bushy cells receive inputs from auditory nerve through large synaptic terminals (calyx of Held), have primary-like responses and accurate temporal coding.
As mentioned earlier, spherical bushy cells and globular bushy cells differ in their
locations and their projections. Spherical cells are located in the more anterior region, and project to MSO, while globular cells project to LSO and trapezoid body.
Spherical cells have one or a few short dendrites that terminate in a dense bush-like
structure near the soma, and globulars have more ovoid somata and larger, more
diffuse dendritic trees. Spherical cells also have lower characteristic frequencies, better phase-locking and are specialized for accurate encoding of AN signal. Globular
bushy cells sometimes chop or have onset responses, receive more and smaller endbulbs, and have higher characteristic frequencies. Probably these cells started out
as one type and have diverged with evolution of sensitivity to higher frequencies.
Bushy cells respond to sound with well-timed action potentials. The characteristics of both the synaptic current and the postsynaptic cell properties are made so
that they shape the timing of the response.
Responses to sounds
32
Figure 13A shows typical responses to tones of bushy cells. Large unitary synaptic events (excitatory post-synaptic potentials; EPSPs) from between one and three
endbulbs cause spherical bushy cells to fire whenever the auditory nerve fibers do
(except when the cell is refractory). This one-spike-in, one-spike-out mode of processing means that the responses to sound of spherical bushy cells resemble those of
auditory nerve fibers, and for this reason they are called “primary-like”. Evidence
that primary-like responses reflect a “one-spike-in, one-spike-out” is provided by
their action potential (AP) shapes. The AP is preceded by a pre-potential (reflecting the build up of the endbulb activity) which are almost always followed by the
postsynaptic component of the spike, demonstrating security of the synapse.
Globular bushy cells give similar response, primary-like-with-notch. They differ
from primary in that in the beginning of the response there is a precisely timed peak
followed by a notch. Precise timing is helped by the convergence of large number of
fibers. If a globular cell needs one input to fire, then there is a very high probability
that one of the incoming fibers will activate early on and cause a spike. This initial
peak will be followed by recovery period (refractoriness), generating notch.
Electrical characteristics
As a response to injection of constant depolarizing current bushy cells produce
one to three spikes at the onset and then settle to a slightly depolarized constant
voltage value. This is due to the low-threshold potassium current. It is partially
activated at rest, then during spiking it strongly activates increasing membrane
conductance and thereby shortening the membrane time constant and repolarizing
the cell. To induce firing, the input must be very strong and fast. The short time
constant blocks temporal integration of inputs. Rapid temporal processing permits
bushy cells to preserve information about the stimulus waveform information that
is necessary for sound localization.
Multipolar cells
Multipolar cells have multiple, long dendrites that extend away from the soma
in several directions. These cells are also sometimes referred to as “stellate”. Two
major classes of multipolar cells have been described. The cells of one group, Tmultipolars (planar), have a stellate morphology with dendrites aligned with auditory nerve fibers , suggesting that these cells receive input from a restricted range of
best frequencies. Their axons project through the trapezoid body (hence the ”T”)
to the contralateral IC. Cells of the second group, D-multipolars (radiate), have
dendritic fields that are not aligned with auditory nerve fibers . Their axons project
to the contralateral cochlear nucleus.
Both T- and D-multipolar cells in the VCN serve the role of interneurons through
their axon collaterals. These terminate locally within VCN as well as projecting to
33
A1
B
A2
D1
C
D2
Figure 13: Different response types. A1: Primary-like response (bushy cell). A2: Primary-like
with notch response (bushy cell). B: Chopper response from T-multipolar cell. C: Single-spike
onset response. D1: Pauser response. D2: builder response. In all panels dark black line under
the axes is the tone presentation. Panels A and B are used by permission of The American
Physiological Society from [8]. Panel C is reprinted from [42] with permission from Elsevier.
Panel D is reprinted by permission of VCH Publishers, Inc. from [29].
the deep DCN. T-Multipolars are excitatory and D-multipolars are inhibitory and
glycin-ergic.
Responses to sounds
Figures 13B shows responses characteristic of T-multipolar cells. T-multipolar
cells respond to tones by firing at regular intervals independent of frequency of the
34
tone, a pattern called chopping. The reproducibility of firing gives histograms of
responses to sound a series of characteristic modes that is independent of the fine
structure of the sound. The intervals between nodes are of equal duration and
correspond to the intervals between spikes, with one spike per node. The peaks
are large at the onset of the response because the latency to the first spike is quite
reproducible in chopper neurons; the peaks fade away over the first 20 msec of the
response, as small variations in interspike interval accumulate and spike times in
successive stimulus repetitions diverge. The chopping firing pattern must arise from
the intrinsic properties of the cells themselves, since it does not reflect properties
of the inputs. Moreover, the same pattern is elicited by depolarization with steady
currents (Fig. 14).
D-multipolar cells are broadly tuned and respond best to stimuli like noise but
only weakly to tones. This is due to the fact that D-multipolars receive input from
many auditory nerve fibers on their somata and on dendrites that spread across
the frequencies. D-multipolars also respond with a precisely timed onset spike to
tones, but (unlike octopus cells) they give some steady discharge after the onset
spike. Also, unlike octopus cells, the firing pattern in response to tones is shaped
by inhibition along with the intrinsic electrical properties.
Electrical characteristics
Unlike bushy cells, multipolars are capable of temporal summation of successive
EPSPs. Recall from above, that bushy and multipolar cells have AMPA receptors
with similar rapid kinetics. The differences between unitary voltage responses arise
because the decay of the unitary event, and therefore the degree of temporal integration, is determined by the membrane time constant of the cell. For bushy cells
this is short (2-4 msec), whereas for multipolar cells it is longer (5-10 msec).
Octopus cells
Octopus cells in VCN (Fig. 11) are contacted by short collaterals of large numbers of auditory nerve fibers through small terminal boutons. Octopus cell dendrites
are oriented in one direction, inspiring their name. The orientation is perpendicular
to the auditory nerve fibers so that the cell bodies encounter fibers with the lowest
best frequencies, and the long dendrites extend toward fibers that encode higher
frequencies. Each cell receives input from roughly one-third of the tonotopic range,
i.e. the frequency tuning is broad and contribution of each fiber to the octopus cell
response is very small. Also EPSPs that an octopus cell receives are very brief,
between 1 and 2 msec in duration.
Responses to sounds
35
Octopus cells behave like bushy cells in that the low input resistance prevents
temporal summation of inputs. But instead of receiving a few large inputs, octopus cells receive small inputs from many fibers. Thus, many fibers have to fire
simultaneously to drive an octopus cell.
Such synchronous firing is not achieved with ongoing pure tone stimuli. Rather,
it happens at stimulus transients. For example, at the onset of a pure tone (thus the
term onset response, Fig. 13C), or at a rapid fluctuation of a broadband stimulus
such as at the onset of a syllable or during a train of clicks. In fact an octopus cell
can faithfully follow a train of clicks with rates up to 500 Hz.
Electrical characteristics
In the case of octopus cells, the fast membrane time constant or high membrane
conductance is generated by presence of two opposing voltage-sensitive currents.
One of them activates with hyperpolarization and has a reversal potential near -40
mV, the other is a potassium current (reversal potential near -80 mV) that activates
with depolarization. At rest both currents are partially activated. In fact, they
are quite large, but they compensate one another. In addition, the voltage-gated
conductances of these currents are very sensitive to voltage near resting potential.
This means that any voltage fluctuation activates these currents which counter the
voltage change. As a result, EPSPs are always small and brief and the cell is only
sensitive to synchronous inputs. Moreover, any depolarizing current that rises too
slowly is bound to activate the potassium conductance and to preclude the cell from
responding. This mechanism makes octopus cells sensitive to the rate of rise of their
input, not its amplitude. In a sense they are sensors of the derivative of the input.
Pyramidal cells
Pyramidal (also called “fusiform”) neurons in the DCN are bipolar, with a spiny
dendritic tree in the molecular layer and a smooth dendritic tree in the deep layer.
The cell bodies of pyramidal cells form a band in the pyramidal cell layer. The
smooth dendrites are flattened in the plane of the isofrequency sheets, where they
receive input from the auditory nerve. The spiny dendrites span the molecular layer
and are contacted by parallel fibers at the spines.
Responses to sounds
Pyramidal neurons in the DCN show pauser and buildup responses to sound.
The examples shown in Fig. 13D are typical of anesthetized animals, where the
inhibitory circuits of the DCN are weakened. The response shows a poorly timed,
long latency onset spike followed by a prominent pause or a slow buildup in response
with a long latency.
Electrical properties
36
Figure 14: Responses of T-multipolar cell to intracellular injections of depolarizing and hyperpolarizing currents. Membrane potential (top) and current time courses (bottom). Spiking is in
response to the depolarizing current. Reproduced with permission from [59], Copyright 1991 by
the Society for Neuroscience.
The pauser and buildup characteristics seem to derive from a transient potassium
conductance. This conductance is inactivated at rest. If a sufficient depolarizing
stimulus arrives, the cell simply responds with a spike. However, if the cell is initially
hyperpolarized, this removes the potassium current inactivation. Any subsequent
depolarization will activate the potassium current transiently, producing the long
latency of a pauser or buildup response. In vivo a sufficient hyperpolarization isprovided by inhibitory synaptic inputs as an aftereffect of a strong response to an
acoustic stimulus.
Some of the other cell types
Giant cells are large multipolar cells located in the deep layers of the DCN.
They have large, sparsely branching, dendritic trees that cross isofrequency sheets.
Giant-cell axons project to the contralateral inferior colliculus.
Granule cells are microneurons whose axons, the parallel fibers, provide a major
excitatory input to DCN through the molecular layer. Granule-cell axons terminate
on spines of the dendrites of pyramidal cells, on spines of cartwheel cells, and on the
stellate cells.
The vertical cells are inhibitory interneurons that project to their isofrequency
sheets in both DCN and VCN; they inhibit all of the principal cells in the cochlear
nucleus, except the octopus cells. Vertical cells are narrowly tuned, and they respond
most strongly to tones at a frequency near their characteristic frequency.
Cartwheel cells are inhibitory interneurons whose numerous cell bodies lie in the
pyramidal cell layer of the DCN. Their dendrites span the molecular layer and are
37
A
B
Figure 15: Weak (A) and strong (B) responses of cartwheel cells. Used by permission of The
American Physiological Society from [71].
densely covered with spines that are contacted by parallel fibers. They contact pyramidal, giant, and other cartwheel cells through glycinergic synapses. The reversal
potential of the cartwheel-to-cartwheel cell synapses lies a few millivolts above the
resting potential and below the threshold for firing so that its effect is depolarizing
for the cell at rest but hyperpolarizing when the cell has been depolarized by other
inputs. Thus, the effects of the cartwheel cell network are likely to be contextdependent: excitatory when the cells are at rest but stabilizing when the cells are
excited.
Figure 15 shows responses to sound of cartwheel cells. Cartwheel cells are the
only cells in the cochlear nucleus with complex action potentials, which reflect a
combined calcium and sodium spike. Many of the cartwheel cells respond weakly to
sounds and no particular pattern of response is consistently observed. They probably
mainly transmit non-auditory information to the principal cells of the DCN.
As a side note: cartwheel cells share many features with cerebellar Purkinje cells.
For example, both of these fire complex action potentials; and genetic mutations
affect Purkinje cells and cartwheel cells similarly.
4.6
Modeling of the cochlear nuclei
As data on the membrane properties of cochlear nucleus neurons have accumulated,
it has become possible to use biophysical models of Hodgkin-Huxley type to explore the different behaviors described in the previous section. These models give
researchers ability to test whether the biophysical mechanisms discussed above can
38
indeed underlie the observed cellular responses. In particular, when many ionic currents have been identified in a given cell type, a model helps identify which of them
are the key ingredients in shaping each feature of the cell’s behavior.
For example, a model of bushy cells [85] demonstrates that the presence of a
low-threshold potassium current in the soma of the cell can account for responses
of both spherical and globular bushy cells depending on number and strength of
inputs. When inputs are few and large the response matches characteristics of the
primary-like response, but with larger number of inputs it becomes primary-like with
notch. Another computational study of globular bushy cells [47] uses a much simpler underlying mathematical model (integrate-and-fire rather than Hodgkin-Huxley
type). This allows to study in generality what increases or decreases synchrony in
the output vs. the input, using globular bushy cell as an example.
Some of the other models have dealt in detail with responses of various cells in
the dorsal cochlear nucleus [9, 80].
Another example of mathematical modeling in cochlear nucleus is a model of
multipolar cell [6]. This is a more complex multi-compartmental model, which
demonstrates which combination of known somatic and dendritic currents can accurately reproduce properties of a chopper response (regular firing with irregular or
constant inputs).
5
Superior olive. Sound localization, Jeffress model
The superior olive is a cellular complex in the brainstem of about 4 mm long. The
cytoarchitecture of this complex defines three parts: the medial superior olive, the
lateral superior olive, and the nucleus of the trapezoid body. It should be emphasized
that the trapezoid body itself is not a nucleus; it is a bundle of fibers. The entire
superior olive complex is surrounded by small cellular groups known as the preolivary
or periolivary nuclei. The olivary nuclear complex is the first level in the auditory
system where binaural integration of auditory signals occur. It is the key station to
performing many of the binaural computations, including, in large part, localization
of sounds.
Two nuclei within superior olivary complex that have been most extensively
studied are the medial and lateral superior olivary nuclei (MSO and LSO). Other
nuclei, such as medial and lateral nuclei of the trapezoid body (MNTB and LNTB)
are named according to their position with respect to the MSO and LSO. There
is also a group of cells called olivacochlear neurons. They project to the cochlea.
These neurons can be activated by sound, and cause suppression of spontaneous and
tone evoked activity in auditory nerves.
39
5.1
Medial nucleus of the trapezoid body (MNTB)
The cell body of neurons in the MNTB receives input from axonal projection of
cells in the contralateral VCN, primarily from the globular bushy cells (GBC). The
GBC-to-MNTB synapses are large, endbulb type, i.e. they also provide reliable
synaptic connection. For that reason the responses of many MNTB cells are similar
to their primary excitatory input, the globular bushy cells. Some cells with chopper
type responses are also found. Because convergence of the bilateral auditory information has not occurred yet at the level of MNTB, the neurons there are responsive
exclusively to sounds presented to the ear contralateral to the nucleus itself. As
usual, each cell responds best to a characteristic frequency. They then send a short
axon to the LSO where it forms inhibitory glycine-ergic synapses.
5.2
Lateral superior olivary nucleus (LSO)
Principal cells receive inputs onto soma and proximal dendrites. Excitatory input is
received directly from the ipsilateral VCN. Inhibitory input is from the contralateral
VCN, transmitted through the MNTB. Even though there is an extra synapse in the
path of the inhibitory signal, inputs from each ear arrive at the LSO simultaneously.
This is possible due to the large reliable synapses associated with MNTB. The cells
that receive excitatory input from ipsilateral ear and inhibitory input from the other
ear are often called IE cells.
Because of the organization of their inputs, LSO cells are excited by the sound
which is louder in the ipsilateral ear and softer in the contralateral ear. There is
almost no response when a sound is louder in the contralateral ear than in the
ipsilateral ear. As a result the sensitivity functions of LSO neurons to the interaural
level difference (ILD) are sigmoidal. There are successful computational models by
Michael Reed and his colleagues (e.g. [81]) that demonstrate how a spectrum of
sound ILDs can be coded by a population of the sigmoidally-tuned LSO cells.
The characteristic frequencies of LSO cells are predominantly in the high frequency range complimenting the range of frequencies over which the MSO is responsive. The frequency tuning is sharp, as narrow as in the cochlear nucleus and
the auditory nerve, because ipsilateral and contralateral inputs are well matched
in frequency. However, ipsilateral (excitatory) inputs are slightly broader tuned.
Therefore, to maintain a response level of an LSO neuron, as contralateral sound
moves away from characteristic frequency it must become louder to continue counteracting the ipsilateral signal.
40
5.3
Medial superior olivary nucleus (MSO)
The cell bodies of neurons in the MSO receive input from two sets of dendrites.
One projects laterally from the cell and gets its input from the ipsilateral VCN. The
other projects medially from the cell and receives its input from the contralateral
VCN. The cells in the MSO, as in LSO, are classified according to their response
to these two sets of dendritic inputs. If a cell is excited by both contralateral and
ipsilateral input, it is classified as an EE cell. If a cell is excited contralateral, but
inhibited by ipsilateral input, it is classified as an EI cell. If a cell is excited by
ipsilateral, but inhibited by contralateral input, it is classified as an IE cell. If a
cell is inhibited by both contralateral and ipsilateral input, that is it fires below its
spontaneous rate when either ear is presented with an appropriate stimulus, it is
classified as an II cell.
MSO neurons respond to both binaural and monaural stimuli. The monaural
response however will always be submaximal when compared to similar binaural
stimuli. The response tends to be frequency selective and the best frequency is
typically the same for both ipsilateral and contralateral input in a given cell. The
characteristic frequencies in the MSO tend to be in the low end of the hearing
spectrum. In addition to a characteristic frequency, many neurons in the MSO
respond best to a specific delay between ipsilateral and contralateral stimulus termed
the “characteristic delay”. They also have a “least characteristic delay” to which
they respond worse than to monaural input. If the characteristic delay is preserved
across frequencies or modulation frequencies, the cell is termed to be of “peaktype”. If the least characteristic delay is preserved across frequencies the cell is
termed “trough-type”. Most of the cells in the MSO are of peak-type, and they
form the basis for a famous Jeffress theory, which is described below.
5.4
Sound localization. Coincidence detector model
Many animals, especially those that function in darkness, rely on sound localization
for finding their prey, avoiding predators, and locating members of their own species.
Therefore, it is not surprising that many mechanisms for sound localization, wellsuited for different circumstances, have been created in the course of evolution.
Primary cues for sound localization are binaural. They include interaural differences in sound pressure level, time of arrival and frequency spectrum of the sound.
It is also possible to localize sounds monaurally, but it is much less efficient (see,
e.g., [106]). Historically, the most studied binaural cues have been interaural time
and sound pressure level differences. Interaural time difference (ITD) arises because
the sound, originating on one side of the head, must follow a longer path and consequently takes a longer time traveling to the distal than proximal ear (Fig. 16).
41
Static
sound source
o
IPD=−30
Neural response
vs.
IPD
o
IPD=45
−30
45
Figure 16: When a sound source is placed not directly in front of the animal, but off-center, it
generates ITD (and IPD). Two schematic sound source locations are shown in the left panel. If
the response of a neuron is recorded at the same time, then it can be plotted vs. corresponding
IPD (schematically shown in the right panel). This plot of response vs. IPD is the static tuning
curve of the neuron.
The interaural level difference (ILD) results from the interference of the head and
ears of the listener with the propagation of the sound so that a “sound shadow”
may develop, i.e., the sound may be attenuated on the far side relative to the near
one. In natural settings an acoustic signal usually gives rise to both time and amplitude differences. Under certain conditions only one of the cues retains sufficient
magnitude to serve the listener. In particular, according to the duplex theory [79],
interaural level differences are the primary cue for localizing high-frequency tones,
whereas interaural time differences are responsible for the localization of the low
frequency tones. The basis for this distinction is that for high-frequency sounds
the wavelength is small compared with the size of the head and the head becomes
an obstacle. At lower frequency sounds the wave length is longer and the head is
acoustically transparent.
In recent years many departures from the duplex theory have been found. For
example, for complex high-frequency signals ITD between low-frequency envelopes
can serve as a localization cue [115]; direct physical measurement of ILD at the ears
of the listener (e.g., in cats [108] and humans [21] show that ILD does not vary in a
systematic fashion with azimuth; further, the interaural differences can be altered by
factors such as head and pinnae shape; finally, interaural spectral differences, that
develop for complex acoustic signals (particularly with high frequency components),
provide considerable directional information that can enhance the localization task.
Nevertheless, time and level differences continue to be considered as primary
42
sound localization cues. Many species are very acutely tuned to them. Humans, for
example, can readily sense interaural disparities of only a few decibels or only tens
of microseconds [79, 102]. The sensitivity is even higher, for example, in some avians
[46]. Besides psychophysical evidence, there is a great amount of physiological data,
showing that in various structures along the auditory pathway (medial and lateral
superior olive (MSO and LSO), dorsal nucleus of lateral lemniscus (DNLL), inferior
colliculus (IC), auditory cortex) low-frequency neurons are sensitive to ITDs and
high-frequency neurons to ILDs. What frequency is “low” or “high” varies with
species. Humans are sensitive to ITDs for tone frequencies up to about 1500 Hz.
Neural sensitivity to ITD in cats, for example, goes up to 3kHz [50], and in owls,
up to 6-8kHz [65].
Interaural phase difference (IPD)
Rose et al. [83] found that for some cells in the inferior colliculus of the cat the
discharge rate was a periodic function of interaural time delay, with a period equal
to the stimulus wavelength. They argued that the periodic nature of the response
showed that the cells were sensitive to the interaural phase difference (IPD) rather
than ITD. Similar cells were found in other parts of the mammalian auditory system
(e.g., [30]). Of course, the issue of distinction between IPD and ITD arises only when
the sound carrier frequency is changed, because, for example, if the cell is sensitive to
ITD, then its preferred IPD (the one that elicits the largest response) would change
linearly with carrier frequency; if, on the other hand, the cell is sensitive to IPD,
its preferred IPD would stay the same. For pure tones of a fixed frequency, talking
about IPD and ITD is redundant and we will sometimes use them interchangably,
assuming that the stimulus is a pure tone at the characteristic frequency of the
neuron.
Schematic of the experiments
In experimental studies concerning the role of binaural cues and corresponding
neuronal sensitivity, the sound is usually played from speakers located at one of
the positions around the listener (free-field), or via headphones (dichotic). Headphone stimulation is, of course, an idealized sound representation (for example, the
stimuli in the headphones are perceived as located within the head, [118]). On the
other hand, it may be hard to control the parameters of the stimuli in the freefield studies. For example, the simplest geometrical model of IPD induction, due
to Woodworth [110], that assumes constant speed and circular head shape, does
not match the direct measurements of IPD [48] (Gourevitch [31] notes that for a
400 Hz signal at 15◦ off midline, the observed IPD is about three times greater than
IPD based on the Woodworth’s model). A more detailed model by Kuhn [49] takes
into account physics of sound wave encountering the head as an obstacle. It shows
43
the dependence of IPD on the head size and the sound carrier frequency for humans and provides better fit for the data, but also demonstrates that extracting
the information available to the auditory system in a free-field study is not a trivial
task.
Neuronal mechanisms of IPD-sensitivity: Superior olive
Many contemporary models of binaural interaction are based on the idea due to
Jeffress [38]. He suggested the following model: an array of neurons receives signals
from the two ears through delay lines (axons that run along the array and provide
conduction delay). The key assumption is that each neuron fires maximally when the
signals from both ears arrive to that neuron simultaneously, in other words — when
the conductance delay exactly compensates the IPD. These neurons are referred to
as coincidence detectors. Mathematically, we can also think about their action as
computing the temporal correlation of signals. The Jeffress-like arrangement was
anatomically found in the avian homolog of medial superior olive (the first site of
binaural interaction), the nucleus laminaris [117]. In mammals, recordings from
single cells in MSO confirm that many cells do have the properties of coincidence
detectors [30, 97, 112]. By recording the responses of these neurons to binaurally
presented tones with different IPDs (simulating sounds at different locations around
the head of the animal), one produces the (static) tuning curve (schematic in Fig.
16). It is usually a relatively smooth curve that has a clearly distinguishable peak.
The peak of the binaural response appears when the ITD is approximately equal
(with a different sign) to the difference between times of maximal responses to
monaural signals [30], as expected for coincidence detectors.
In natural environments the relative positions of a sound source and the head are
free to change and, therefore, the binaural sound localization cues are likely to be
changing (dynamic) too. In experiments (e.g., [113]), the binaural cue produced by
a moving sound source may be simulated by temporal variation of IPD. (In humans
dynamic IPD modulation creates the percept of sound motion.) If the MSO cells
act like coincidence detectors, they are expected to signal the instantaneous value of
the interaural delay and show no evidence for motion sensitivity. This is confirmed
experimentally (e.g., [98]).
Coincidence detector model
Cell body
We describe here a coincidence detector point neuron model of the Morris-Lecar
type (after Agmon-Snir et al. [2]). The main dynamic variables are the voltage V
and a gating variable w.
C V̇ = −gN a m∞ (V )(V − VN a ) − gK · w · (V − VK ) − gL (V − VL ) − Gsyn (V − Vsyn )
44
τw (V )ẇ = w∞ − w
Parameters are chosen in a realistic range to tune the model neuron into the phasic
firing regime. Namely, this cell will fire only one spike in response to a depolarizing
current injection (Fig. 17), as observed experimentally (e.g. [82]). Most of the
parameters are the same as in Agmon-Snir et al. [2], unless noted otherwise.
Responses to injected constant current
−15
−20
−25
−30
V, mV
−35
−40
−45
−50
−55
−60
−65
−2
0
2
4
6
8
10
12
t
Figure 17: Responses to increasing level of depolarizing current. The cell fires only once in
response to a constant level injection of depolarizing superthreshold current.
Input
Input is modeled as 2 delay lines (one from each ear). Each line makes 12
synapses on the cell (Fig. 18A). On each stimulus cycle every synapse can produce
a synaptic event with probability 0.7 (at 500 Hz stimulus frequency this corresponds
to average firing rate of the afferents equal to 350 Hz) (Fig. 18B). In case a synaptic
event is produced, it results in the alpha-function change of postsynaptic conductance, namely
Ã
!
t−φ
t−φ
exp 1 −
.
Gsyn,unit = A
tp
tp
Here, tp = .1 msec, amplitude A is chosen to produce reasonable firing rates at the
given number of inputs and φ is chosen from a Gaussian distribution N (φ0 , σ 2 ) for
one side and N (IPD, σ 2 ) for the other side (IPD – interaural phase disparity, σ = 0.4
– measure of phase-locking, φ0 – preferred phase) (see Fig. 19A).
Figures 19 and 20 show sample voltage and synaptic conductance time courses
and a tuning curve of the model neuron. The tuning curve was obtained by computing number of spike elicited during a 1 sec stimulus presentation. This tuning
curve shows that the neuron is a pretty good coincidence detector.
45
A
B
0
1
1
0
1 0
0
01
1
1
0
1
0
0
1
Line 1
# inputs
00
11
1
0
0
00
11
00
01
1
0
1
0011
11
Line 2
# inputs
time
Figure 18: Schematic of the input to the model cell. A: Delay lines. B: Example of distribution
of input events times within stimulus cycle. Total number of events is also random, see text
A
B
Example of voltage time course
Example of synaptic conductance on 1 line
−25
25
−30
20
−35
−40
15
Gsyn
V, mV
−45
−50
10
−55
−60
5
−65
0
−2
0
2
4
6
8
10
−70
−2
12
0
2
4
6
8
10
12
Time, msec
Time, msec
Figure 19: Sample conductance and voltage traces. A: Synaptic conductance of one line of
inputs over 6 stimulus cycles. B: Corresponding voltage trace.
However, the known coincident-detector avian auditory neurons have a very specific structure. They have bipolar dendrites whose length varies continuously with
characteristic frequency of the neuron (neurons with higher characteristic frequencies have shorter dendrites). The function of the dendrites was hard to unravel
experimentally, until a mathematical model [2] suggested that propagation of signal
along the dendrites resulted in nonlinear summation of conductances at the soma,
improving coincidence detection. The argument goes as follows. Let us measure the
quality of coincidence detection as the difference between maximum and minimum
of the tuning curve. Let us also assume that the unitary synaptic conductance is
adjusted in such a way that when signals from both sides arrive simultaneously the
probability of firing is the same for either a point neuron or a bipolar neuron (this
46
Tuning curve
220
200
Firing rate (spikes/sec)
180
160
140
120
100
80
60
−300
−200
−100
0
IPD
100
200
300
Figure 20: Tuning curve of the model neuron. It is computed with small amount of averaging.
ensures that the maximum of the tuning curve is the same), i.e. let us say that
the cell needs n simultaneous input spikes to fire (n/2 from each side). First, we
need to show that the quality of coincidence detection is better for a neuron with
bipolar dendrite than for a point neuron (i.e. minimum of the tuning curve is lower
with the dendrite included). The key observation is that the synaptic current produced from a number of input spikes arriving at the same location is smaller than
the current produced by the same number of inputs arriving at several different
locations, because the driving force for excitatory current decreases with increase in
local voltage. When the signals from different sides arrive half-cycle apart (worst
ITD) and by chance one side produces n input spikes, it is still enough to fire a
point neuron (which does not care where the inputs came from), but in the bipolar
dendrite neuron all n spikes will now be concentrated on one side, producing smaller
(insufficient) synaptic current, i.e. probability of firing will be smaller with dendrite
included. Second, we need to show that longer dendrites improve coincidence detection at lower rather than higher input frequencies. This happens because of the
jitter present in the input trains. Higher frequencies mean larger overlap between
inputs from both sides even when the phase shift is maximal. These occasional
“false” coincidences, will only add a small fraction to the conductance. For the
point neuron it means only a small change in synaptic current. However, in case
with dendrite the small conductance arrives at the different dendrite, producing relatively large current, i.e. higher probability of the “false” response. As a result, at
a given level of jitter, the minimum of the tuning curve increases with frequency,
i.e. the dendrites become less advantageous at higher frequencies.
In addition, more recent computational models of MSO argue that Jeffress model
47
may need to be seriously revised. In fact, it has been shown experimentally that
MSO cells that ate ITD-sensitive may be receiving substantial location-tuned inhibitory input [13]. A computational model [13] confirms that if that inhibition
arrives at the MSO cell with a fixed time delay relative to excitation from the same
side of the brain, it may allow to explain a larger range of MSO response properties
than traditional excitation-only coincidence detector Jeffress-type model.
6
Midbrain
The inferior colliculus (IC) is the midbrain target of all ascending auditory information. It has two major divisions, the central nucleus and dorsal cortex, and both are
tonotopically organized. The inputs from brainstem auditory nuclei create multiple
tonotopic maps to form what are believed to be locally segregated functional zones
for processing of different aspects of the auditory stimuli. The central nucleus receives both direct monaural input from cochlear nuclei and indirect binaural input
from the superior olive.
6.1
Cellular organization and physiology of mammalian IC
The IC appears to be an auditory integrative station as well as a switchboard.
It is responsive to interaural delay and amplitude differences and may provide a
spatial map of the auditory environment, although this has not been directly shown,
except in some birds, e.g., barn owls. Changes in the spectrum of sounds such as
amplitude and frequency modulation appear to have a separate representation in
the IC. This sensitivity to spectral changes possibly provides the building blocks
for neurons responsive to specific phonemes and intonations necessary to recognize
speech. Finally, the IC is involved in integration and routing of multi-modal sensory
perception. In particular, it sends projections which are involved in ocular reflexes
and coordination between auditory and visual systems. It also modifies activity in
regions of the brain responsible for attention and learning.
One of the very interesting properties of the inferior colliculus responses is its
sensitivity to interaural time (or phase) differences. As we explained above, ITD
sensitivity first arises in the MSO, due to precise preservation of timing information
(phase locking) and coincidence detection. Notably, preservation of timing information starts to deteriorate in the IC. In particular even its low frequency cells do not
phase lock as well as cells at the previous levels. Yet, in spite of apparent lack of timing information, new properties of ITD sensitivity, such as non-linear responses to
dynamic changes in ITD arise in the inferior colliculus. This has recently attracted
attention of modelers to the IC [14, 15, 88, 11, 12].
48
6.2
Modeling of the IPD sensitivity in the inferior colliculus
The inferior colliculus (IC), in the midbrain, is a crucial structure in the central
auditory pathway (Fig. 1). Ascending projections from many auditory brain stem
nuclei converge there. Nevertheless, its role in the processing of auditory information
is not well understood. Recent studies have revealed that many neurons in the IC
respond differently to auditory stimuli presented under static or dynamic conditions
and the response also depends on the history of stimulation. These include responses
to dynamically varying binaural cues: interaural phase [62, 95, 96] and interaural
level differences [87], and monaural stimuli such as slowly modulated frequency [57].
At least some of the sensitivity to dynamic temporal features is shown to originate
in IC or in the projections to IC, because, as we mentioned above, responses of cells
in lower level structures (MSO and dorsal nuclei of lateral lemniscus (DNLL) for the
low-frequency pathway, Fig. 1) follow the instantaneous value of the stimulus [98].
A history-dependence of the responses, similar to that in IC, but of even greater
magnitude, has been observed in the auditory cortex [58]. These findings suggest
that there might be a hierarchy of representations of auditory signals at subsequent
levels of the auditory system.
Sensitivity to simulated motion in IC has been demonstrated with binaural beats
(periodic change of IPD through 360◦ ), [95, 114] and partial range sweeps (periodic
modulation of interaural phase in a limited range) [96, 62]. For example, it was
shown that the binaural beat response is sometimes shifted with respect to the
static tuning curve (Fig. 22, 23), responses to IPD sweeps form hysteresis loops
(Fig. 24), and the response to the dynamic stimulus can be strong outside of the
statically determined excitatory receptive field (example in Fig. 24A, arrow). Thus,
there is a transformation of response properties between MSO and IC, but there is
no agreement on how this transformation is achieved. Both intrinsic and synaptic
mechanisms have been implicated.
One hypothesis is that the observed history-dependence of responses is reflective of “adaptation of excitation” (firing rate adaptation) [62, 98]. The firing rate
adaptation is a slow negative feedback mechanism that can have various underlying
cellular or network mechanisms. It can manifest itself in extracellular recordings of
responses to a constant stimulus as the decrease of probability of firing with time,
and in in vitro recordings as the increase of the interspike interval in response to
the injection of a constant depolarizing current. Both extracellular [62, 98], and
intracellular [72, 93] recordings have registered firing rate adaptation in IC. It has
also been shown by Cai et al., in a modeling study [15], that addition of a negative
feedback, in the form of a calcium-activated hyperpolarizing membrane conductance
(one of the adaptation mechanisms), to a motion-insensitive model of an IC neuron,
induced sensitivity to dynamic stimuli.
49
Another hypothesis [98] is that the motion is the result of interaction between
excitatory and inhibitory inputs, aided by adaptation and post-inhibitory rebound.
Post-inhibitory rebound (PIR) is observed when the neuron fires upon its release
from hyperpolarization; this has been demonstrated in IC slice recordings [72, 93].
McAlpine and Palmer [64] argued that leaving the key role in generation of motion
sensitivity to inhibition contradicts their data. They showed that sensitivity to
the apparent motion cues is decreased in the presence of the inhibitory transmitter,
GABA, and increased in the presence of inhibitory transmission blocker, bicuculline.
Unification of the hypotheses. To examine the role of various mechanisms in
shaping IC response properties, mathematical models of an IC cell were developed
in which the cell receives IPD-tuned excitatory and inhibitory inputs and possesses
the properties of adaptation and post-inhibitory rebound.
In earlier modeling studies of IC, Cai et al. [14, 15] developed detailed cell-based
spike-generating models, involving multiple stages of auditory processing. One of
their goals was to describe some of the data that require a hierarchy of binaural
neurons. Recent models [11, 12] take a very different approach. These models are
minimal in that they only involve the components whose role we want to test and do
not explicitly include spikes (firing-rate-type models). Exclusion of spikes from the
model is based on the following consideration. First, IC cells, unlike neurons in MSO,
do not phase lock to the carrier frequency of the stimulus. In other words, spike
timing in IC is less precise. Second, it is assumed that the convergent afferents from
MSO represent a distribution of preferred phases and phase-locking properties. As a
result, either through filtering by the IC cell or dispersion among the inputs, spiketiming information is not of critical importance to these IC cells. This minimalistic
approach greatly facilitates the examination of computations that can be performed
with rate-coded inputs (spike-timing precision on the order of 10 ms), consistent with
the type of input information available to neurons at higher levels of processing. In
addition, these models are computationally efficient, can be easily implemented, and
require minimal assumptions about the underlying neural system.
It is shown in the modeling studies [11, 12] that intrinsic cellular mechanisms,
such as adaptation and post-inhibitory rebound, together with the interaction of
excitatory and inhibitory inputs, contribute to the shaping of dynamic responses.
It is also shown that each of these mechanisms has a specific influence on response
features. Therefore, if any of them is left out of consideration, the remaining two
cannot adequately account for all experimental data.
Model
Cellular properties
The model cell [12] represents a low-frequency-tuned IC neuron that responds
50
to interaural phase cues. Since phase-locking to the carrier frequency among such
cells is infrequent and not tight [51], this model assumes that binaural input is
rate-encoded. In this model the spikes are eliminated by implicitly averaging over
a short time scale (say, 10 ms). Here, we take as the cell’s observable state variable
its averaged voltage relative to rest (V ). This formulation is in the spirit of early
rate models (e.g., [32]), but extended to include intrinsic cellular mechanisms such
as adaptation and rebound. Parameter values are not optimized, most of them are
chosen to have representative values within physiological constraints, as specified in
this section.
The current-balance equation for the IC cell is:
C V̇ = −gL (V − VL ) − ḡa a(V − Va ) − Isyn (t) − IP IR + I.
(16)
The terms on the right-hand side of the equation represent (in this order) the leakage,
adaptation, synaptic, post inhibitory rebound, and applied currents. The applied
current I is zero for the examples considered here and all other terms are defined
below.
Leakage current has reversal potential VL = 0 mV and conductance gL = 0.2
mS/cm2 . Thus, with C = 1 µF/cm2 , the membrane time constant τV = C/gL is 5
ms.
The adaptation is modeled by a slowly-activating voltage-gated potassium current ḡa a(V − Va ). Its gating variable a satisfies
τa ȧ = a∞ (V ) − a.
(17)
Here a∞ (V ) = 1/(1 + exp(−(V − θa )/ka )), time constant τa = 150 ms, ka = 5 mV,
θa = 30 mV, maximal conductance ḡa = 0.4 mS/cm2 , reversal potential Va = −30
mV. The parameters ka , θa are chosen so that little adaptation occurs below firing
threshold; and ḡa so that when fully adapted for large inputs, the firing rate is
reduced by 67%. The value of τa , assumed to be voltage-independent, matches
the adaptation time scale range seen in spike trains (Semple, unpublished) and it
leads to dynamic effects that provide good comparison with results over the range
of stimulus rates used in experiments.
The model’s readout variable, r, represents firing rate, normalized by an arbitrary constant. It is an instantaneous threshold-linear function of V : r = 0 for
V < Vth and r = K · (V − Vth ) for V ≥ Vth , where Vth = 10 mV. We set (as in
[12]) K = 0.04075 mV−1 . Its value does not influence the cell’s responsiveness, only
setting the scale for r.
Synaptic inputs
As proposed by Spitzer and Semple [98] and implemented in the spike-based
model of Cai et al. [14, 15], this model IC cell receives binaural excitatory input
51
from neurons in ipsilateral MSO (e.g., [1, 70]) and indirect (via DNLL [91]) binaural inhibitory input from the contralateral MSO. Thus DNLL, although it is not
specifically included in the model, is assumed (as in Cai et al. [14]) to serve as an
instantaneous converter of excitation to inhibition.
The afferent input from MSO is assumed to be tonotopically organized and
IPD-tuned. Typical tuning curves of MSO neurons are approximately sinusoidal
with preferred (maximum) phases in contralateral space [97, 112]. We use a sign
convention that the IPD is positive if the phase is leading in the ear contralateral
to the IC cell, to choose the preferred phases φE = 40◦ for excitation and φI =
−100◦ for inhibition. Preferred phases of MSO cells are broadly distributed [63, 58].
Similar ranges work in the model, and in that sense, the particular values picked
are arbitrary.
While individual MSO cells are phase-locked to the carrier frequency [97, 112],
we assume that the effective input to an IC neuron is rate-encoded, as explained
above. Under this assumption, the synaptic conductance transients from incoming
MSO spikes are smeared into a smooth time course that traces the tuning curve of
the respective presynaptic population (Fig. 21). Therefore, the smoothed synaptic
conductances, averaged over input lines and short time scales, gE and gI , are proportional to the firing rates of the respective MSO populations and they depend on
t if IPD is dynamic. Then
Isyn = −gE (V − VE ) − gI (V − VI ),
(18)
gE,I = ḡE,I [0.55 + 0.45 cos(IP D(t) − φE,I )] .
(19)
Reversal potentials are VE = 100 mV, VI = −30 mV, maximum conductance values
ḡE = 0.3 mS/cm2 , ḡI = 0.43 mS/cm2 .
Rebound mechanism
The post-inhibitory rebound (PIR) mechanism is implemented as a transient
inward current (IP IR in the equation (16)):
IP IR = gP IR · m · h · (V − VP IR ).
(20)
The current’s gating variables, fast (instantaneous) activation, m, and slow inactivation, h, satisfy:
m = m∞ (V ),
τh ḣ = h∞ (V ) − h,
(21)
(22)
where m∞ (V ) = 1/(1+e−(V −θm )/km ), h∞ (V ) = 1−1/(1+e−(V −θh )/kh ), τh = 150 ms
and VP IR = 100 mV. The parameter values: km = 4.55 mV, θm = 9 mV, kh = .11
52
IC
+
Tuning
g
IPD
DNLL
Tuning
E
−180
−
MSO
MSO
IPSI
CONTRA
180
gI
−180
IPD
180
Figure 21: Schematic of the model. A neuron in the inferior colliculus (marked IC), receiving
excitatory input from ipsilateral medial superior olive (MSO); contralateral input is inhibitory.
Each of the MSO-labelled units represents an average over several MSO cells with similar response
properties (see text). DNLL (dorsal nucleus of lateral lemniscus) is a relay nucleus, not included
in the model. Interaural phase difference (IPD) is the parameter whose value at any time defines
the stimulus. The tuning curves show the stimulus (IPD) tuning of the MSO populations. They
also represent dependence of synaptic conductances (gE and gI ) on IPD (see text). Used by
permission of The American Physiological Society from [12].
mV, θh = −11 mV, are chosen to provide only small current at steady state for
any constant V and maximum responsiveness for voltages near rest. Maximum
conductance gP IR equals 0.35 mS/cm2 in Fig. 24 and zero elsewhere.
Stimuli
The only sound input parameter that is varied is IPD as represented in equations (18) and (19); the sound’s carrier frequency and pressure level are fixed. The
static tuning curve is generated by presentation of constant-IPD stimuli. The dynamic stimuli are chosen from two classes. First, the binaural beat stimulus is
generated as: IP D(t) = 360◦ · fb · t (mod 360◦ ), with beat frequency fb that can
be negative or positive. Second, the partial range sweep is generated by IP D(t) =
Pc + Pd · triang(Pr t/360). Here, triang(·) is a periodic function (period 1) defined
by triang(x) = 4x − 1 for 0 ≤ x < 1/2 and triang(x) = −4x + 3 for 1/2 ≤ x < 1.
The stimulus parameters are the sweep’s center Pc ; sweep depth Pd (usually 45◦ );
and sweep rate Pr (in degrees per second, usually 360◦ /s). We call the sweep’s
half cycle where IPD increases – the “up-sweep”, and where IPD decreases – the
“down-sweep”.
Simulation data analysis
The response’s relative phase (Figs. 22 and 23) is the mean phase of the response
53
to a binaural beat stimulus minus the mean phase of the static tuning curve. The
mean phase is the direction of the vector that determines the response’s vector
strength [30]. To compute the mean phase we collect a number of response values
(rj ) and corresponding stimulus values (IP Dj in radians). The average phase ϕ is
P
P
such that tan ϕ = ( rj sin(IP Dj ))/( rj cos(IP Dj )). For the static tuning curve
the responses were collected at IPDs ranging from -180◦ to 180◦ in 10◦ intervals. For
the binaural beat the response to the last stimulus cycle is used.
Example of results: binaural beats
The binaural beat stimulus has been used widely in experimental studies (e.g.,
[52, 61, 96, 107] to create interaural phase modulation: a linear sweep through a full
range of phases (Fig. 22). Positive beat frequency means IPD increases during each
cycle; negative beat frequency — IPD decreases (Fig. 22A). Using the linear form
of IPD(t) to represent a binaural beat stimulus in the model, we obtain responses
whose time courses are shown in Fig. 22A, lower.
To compare responses between different beat stimuli and also with the static
response, we plot the instantaneous firing rate vs. the corresponding IPD. Fig.
22B shows an example that is typical of many experimentally recorded responses
(e.g., [52, 95, 96]). Replotting our computed response from 27A in 27C reveals
features in accord with the experimental data (compare 27B and 27C): dynamic
IPD modulation increases the sharpness of tuning; preferred phases of responses to
the two directions of the beat are shifted in opposite directions from the static tuning
curve; maximum dynamic responses are significantly higher than maximum static
responses. These features, especially the phase shift, depend on beat frequency (Fig.
22C, D).
Dependence of phase on the beat frequency
In Fig. 22D the direction of phase shift is what one would expect from a lag
in response. Fig. 22C is what would be expected in the presence of adaptation
(e.g., [94]; as the firing rate rises, the response is “chopped off” by adaptation at the
later part of the cycle. This shifts the average of the response to earlier phases). In
particular, if we interpret the IPD change as an acoustic motion, then for a stimulus
moving in a given direction there can be a phase advance or phase-lag, depending
on the speed of the simulated motion (beat frequency).
To characterize the dependence of the phase shift on parameters, we define the
response phase as in a vector-strength computation (see above). We take the average
phase of the static response as a reference (ϕ0 = 0.1623). Fig. 23A shows the
relative phase of the response (ϕ̃fb − ϕ0 ) vs. beat frequency (fb ). At the smallest
beat frequencies the tuning is close to the static case and the relative phase is
close to zero. As the absolute value of beat frequency increases, so does the phase
54
B
0
200
400
600
800
1000
Normalized
firing rate
−180
1
0
0
1
200
400
600
Time (msec)
800
D
Static
2 Hz
−2 Hz
EXPERIMENT
50
0
−180
1000
1
−90
0
IPD (degrees)
90
180
Static
20 Hz
−20 Hz
Nomalized
firing rate
Normalized
firing rate
C
Static
5 Hz
−5 Hz
100
0
Discharge rate (Hz)
IPD
A 180
0
−100
0
IPD (degrees)
0
100
−100
0
IPD (degrees)
100
Figure 22: Responses to binaural beat stimuli. A: Time courses of IPD stimuli (upper) and
corresponding model IC responses (lower). Beat frequencies 2 Hz (solid) and -2 Hz (dashed).
B-D: Black lines show responses to binaural beats of positive (solid) and negative (dashed)
frequencies. Grey line is the static tuning curve. B: Example of experimentally recorded response.
Carrier frequency 400 Hz, presented at binaural SPL 70 dB. Beat frequency 5 Hz. C,D: Model
responses over the final cycle of stimulation vs corresponding IPD. Beat frequency is equal to 2
and 20 Hz, respectively. Used by permission of The American Physiological Society from [12].
advance. The phase-advance is due to the firing rate adaptation (see above). At yet
higher frequencies a phase lag develops. This reflects the RC-properties of the cell
membrane — the membrane time constant prevents the cell from responding fast
enough to follow the high frequency phase modulation.
Plots such as in Fig. 23A have been published for several different IC neurons
[98, 12] and examples are reproduces in Fig. 23B. Whereas the present model does
not have a goal of quantitative match between the modeling results and experimental data, there are two qualitative differences between Figs. 23A and 23B. First,
the range of values of the phase shift is considerably larger in the experimental
55
Relative phase
A
0
0.1
−0.1
0
2
1
−10
B
0
−1
−10
−10
Beat frequency
−10
2
0
−1
0
10
1
10
10
1
2
10
Stimulus
IPD
Response
(no delay)
1
Relative phase
3
1
0
10
10
Beat frequency
C
0.13
Relative phase (cycles)
−1
10
d 30
Response
(delayed)
10
d
5
0
0
0
−1
10
1
−1
2
10
Beat frequency
10
10
E
0
1
0
1
10
10
Beat frequency
2
10
1
Average normalized
firing rate
50
Discharge rate (Hz)
D
0
10
g
I
0
−1
10
0
1
10
10
Beat frequency
0
−1
10
2
10
10
10
Beat frequency
2
10
Figure 23: A: Dependence of phase shift on beat frequency in the model neuron. Phase shift
is relative to the static tuning curve (see text). B: Experimentally recorded phase (relative to
the phase at 0.1 Hz beat frequency) vs beat frequency. Six different cells are shown. Carrier
frequencies and binaural SPLs are: 1400 Hz, 80 dB (star); 1250 Hz, 80 dB (circle); 200 Hz,
70 dB (filled triangle); 800 Hz, 80 dB (diamond); 900 Hz, 70 dB (empty triangle); 1000 Hz,
80 dB (square). Inset is a zoom-in at low beat frequencies. C: Phase of the model response,
recomputed after accounting for transmission delay from outer ear to IC, for various values of
delay (d), as marked. Inset illustrates computation of delay-adjusted phase. D,E: Average firing
rate as a function of beat frequency: experimental (D, same cells with the same markers as in
B) and in the model (E). Multiple curves in E – for different values of ḡI (ḡI = 0.1, 0.2, 0.3,
0.43, 0.5, 0.6, 0.7, 0.8 mS/cm2 ). Thick line — for the parameter values given in Methods. Used
by permission of The American Physiological Society from [12].
56
recordings, particularly, the phase lag at higher beat frequencies. Second, it had
been reported in the experimental data, that there is no appreciable phase advance
at lower beat frequencies. Fig. 23B suggests that the phase remains approximately
constant before shifting in the positive direction. Only in the close-up view of the
phase at low beat frequencies (Fig. 23B, inset) can the phase-advance be detected.
It is a surprising observation, given that the firing rate, or spike frequency, adaptation has been reported in IC cells on many occasions (e.g., [62, 96]) and that, as
we noted above, the phase advance is a generic feature of cells with adapting spike
trains.
Role of transmission delay
The modeling study [12] suggested that the absence of phase advance and high
values of phase lag should be attributed to the transmission delay (five to tens of
milliseconds) that exists in transmission of the stimulus to the IC. This delay was
not included in the model, as outlined in Model above. In fact, we can find its contribution analytically (Fig. 23C, inset). Consider the response without transmission
delay to the beat of positive frequency fb . In the phase computation, the response at
each of the chosen times (each bin of PSTH) is considered as a vector from the origin
with length proportional to the recorded firing rate and pointing at the corresponding value of IPD. In polar coordinates r̃j = (rj , IP Dj ), where rj is the jth recorded
data point (normalized firing rate), IP Dj is the corresponding IPD. The average
phase of response (ϕ̃fb ) can be found as a phase of the sum of these vectors. If the
response is delayed, then the vectors should be ~rj = (rj , IP Dj + d · fb ·360◦ ), because
the same firing rates would correspond to different IPDs (see Fig. 23C, inset). The
vectors are the same as without transmission delay, just rotated by d · fb fractions of
the cycle. Therefore, the phase with transmission delay ϕfb = ϕ̃fb + d · fb . If we plot
the relative phase with the transmission delay, it will be ϕfb − ϕ0 = (ϕ̃fb − ϕ0 ) + d · fb
The phase of the static tuning curve (ϕ0 ) does not depend on the transmission delay.
Fig. 23C shows the graph from 23A right, modified by various transmission delays
(thick curve is without delay — same as in 23A right). With transmission delay
included in the model, the range of observed phase shifts increases dramatically
and the phase-advance is masked. In practice neurons are expected to have various
transmission delays, which would result in a variety of phase-frequency curves, even
if the cellular parameters were similar.
Mean firing rate
Another interesting property of response to binaural beats is that the mean firing
rate stays approximately constant over a wide range of beat frequencies (Fig. 23D
and [98]). The particular value, however, is different for different cells. The same
is valid for the model system (Fig. 23E). The thick curve is the firing rate for the
57
A
THEORY
Normalized firing rate
EXPERIMENT
Discharge rate (Hz)
100
0
1
B
−50
0
50
100
0
−50
150
IPD
0
50
IPD
100
150
Figure 24: Responses to partial range sweeps. “Rise-from-nowhere”. A: Experimentally observed response arising in silent region of the static tuning curve (marked with an arrow). Carrier
frequency 1050 Hz; binaural SPL 40 Hz. B: Model responses to various sweeps. The rightmost
sweep produces “rise-from-nowhere” (arrow). Used by permission of The American Physiological
Society from [12].
standard parameter values. It is nearly constant across all tested beat frequencies
(0.1 to 100 Hz). The constant value can be modified by change in other parameters
of the system, e.g. ḡI (shown).
Other results
The model that we have just described allows to show [12] that the presence of
firing rate adaptation together with IPD-tuned excitatory and inhibitory inputs can
explain sharper tuning and phase shifts in response to beats; discontiguous responses
to overlapping IPD sweeps; and a decrease in motion sensitivity in the presence
of added inhibition. Also, a post-inhibitory rebound mechanism (modeled as a
transient membrane current activated by release from prolonged hyperpolarization)
allows to explain the strong excitatory dynamic response to sweeps in the silent
portion of the static tuning curve (Fig. 24A,B arrows). It also allows to make
predictions for future in vitro and in vivo experiments and suggest experiments that
might help to clarify what are the sources of inhibition. For further details and
discussion of this model see [12].
7
Thalamus and cortex
The auditory part of the thalamus is the medial geniculate body. It is situated in
the caudal part of the thalamus and is the last major relay station for ascending
auditory fibers before they reach the cortex. There is a tonotopic arrangement in
58
the medial geniculate body in which low frequencies are represented laterally and
high frequencies are located medially in the principal part. The main projection of
the medial geniculate body is to the primary auditory cortex. The medial geniculate
body also sends fibers to other thalamic nuclei and may play a part in a regulatory
feedback system, with descending projections to the inferior colliculus, the nucleus
of the lateral lemniscus, the trapezoid body, and the superior olivary nucleus.
The primary auditory cortex is a primary auditory reception area in cortex and it
receives its projections from the medial geniculate body. Areas immediately adjacent
to the primary auditory cortex are auditory-association areas. These association areas receive signals from the primary auditory cortex and send projections to other
parts of the cortex. Tonotopic organization of the auditory cortex is particularly
complex. In the simplest analysis, high frequencies are represented anteriorly and
low frequencies posteriorly in the auditory cortex. Each auditory cortical area is reciprocally connected to an area of the same type area in the contralateral hemisphere.
In addition, auditory-association areas connect with other sensory-association areas
such as somatosensory and visual. They also send projections that converge in the
parietotemporal language area. It appears that the higher level of integration in the
association areas is responsible for more complex interpretation of sounds. These
properties of the auditory cortex may explain why patients with lesions in one of
the cortical hemispheres have little difficulty with hearing as measured by presenting
simple sounds, e.g., pure tones. However, such patients may have impaired ability
to discriminate the distorted or interrupted speech patterns and have difficulty focusing on one stimulus if a competing message, especially meaningful, i.e., speech is
presented at the same time.
There is also a descending efferent auditory pathway that parallels the afferent
pathway and is influenced by ascending fibers via multiple feedback loops. The specific function of this system in audition is not well understood, but clearly modulates
central processing and regulates the input from peripheral receptors.
Because auditory cortex seems to deal primarily with higher brain functions, its
modeling has been very scarce. However, recently there has been an promising set
of studies by Middlebrooks and colleagues, in which they teach neural networks to
interpret cortical recordings and make predictions of the psychophysical performance
[24]. More details will have to be unravelled about biophysical properties of cortical
auditory cells before more detailed modeling can be successfully used.
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